Contrasting with the Genius Ditz, who, while usually foolish and moronic, has flashes of insight or hidden talents, we have the Ditzy Genius who, while very intelligent and talented, has absolutely nothing in the way of common sense, logic, wit or tact.
This can occasionally be shown as naivete to the point of stupidity, and/or nigh-Suicidal Overconfidence / inability to see the danger in what they're doing. In any case, these characters end up landing themselves in trouble more often than not.
The primary difference between a Ditzy Genius and a Genius Ditz is that a Ditzy Genius has high general intelligence but is highly inept at something more mundane than their intellectual interests (social skills, for example) or lacks common sense, whereas the Genius Ditz usually seems dumb or clueless but is in fact highly proficient in at least one fairly specialised skill or field of knowledge. Contrast also Book Dumb, which is a character smart in mundane life but abysmal in everything academic.
Sometimes this is a deliberate characterization decision made to give a highly intelligent character a Fatal Flaw, or make them come across as weird, or any number of other reasons. Other times it happens entirely by accident because the writers keep tossing the Idiot Ball to a character who's (perhaps only because we're told as much) supposed to be academically brilliant.
See also Cloud Cuckoo Lander or Brilliant, but Lazy, especially in cases where the character's intelligence is largely informed. One of the standard Hollywood interpretations of a Nerd, the other two being the Hollywood Nerd and the Extraverted Nerd. Very closely related to Smarter Than You Look. Also compare Absent-Minded Professor, who isn't foolish so much as forgetful. See also No Social Skills and Sacrificed Basic Skill for Awesome Training.
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Anime & Manga
Bulma from Dragon Ball is a prime example. While a mechanical genius and heir to a huge scientific corporation, she has absolutely no sense at all when it comes to dealing with danger. And yet somehow she gets the idea to become a treasure hunter. She gets only slightly better at this over the course of the series, in the sense that she doesn't walk headlong into danger without thinking first so much.
It runs in the family, because her father, Dr. Briefs, is even worse—he's a genius in anything mechanical, but has absolutely no common sense.
Lloyd in Code Geass. Brilliant mecha designer, does not understand basic human interaction. The characters are shocked to hear that he is engaged. It's an arranged marriage, since Lloyd is a Count.
Kirino Chiba of Bamboo Blade is frequently behaving in a ditzy, overexcited manner. She is also ranked 20th in her year at school. Not genius level, but highly respectable nonetheless.
Lamia Loveless in Super Robot Wars is a completely competent Super Robot pilot - calm, analytical and deadly in battle. During her normal life, however, she tends to act ditzy. This is justified, since she is a Ridiculously Human Robot who was created mere years ago, and had only recently discovered what it felt like to be human.
In Hayate the Combat Butler, Maria and Nagi have both skipped grades and know multiple languages. When they are split up while on a train, Hayate is torn between protecting his master or staying with the maid until Hinagiku shows up for a less-than-fully-heroic rescue because they are completely clueless about the world.
Tenpou in Saiyuki Gaiden brilliant strategist, but seems baffled by everyday life and would probably be buried under a pile of books almost permanently if it wasn't for Kenren.
So much so that one fanfiction diagnosed Heiji with Asperger's Syndrome in a fictionalized psych profile.
Tamaki in Ouran High School Host Club is rather ditzy. He also has the second highest grades in his year without even trying, to the frustration of some.
Harumi Kiyama in A Certain Scientific Railgun is a very intelligent scientist, but lacks even the most basic shred of common sense. For example, her thought process when it's hot essentially comes down to "It's hot, therefore I should take off some clothes." The fact that she's doing this in public doesn't occur to her. She actually becomes the source of an Urban Legend about a "Stripper Lady" because of this.
Black from Pokémon Special. Type match-ups, move accuracy, evolutionary lines, detailed information on every Gym Leader and Elite Four member (including their team line-ups), Black studied them all before going out on his quest To Be a Master. Pity he has No Social Skills and feels that it's appropriate to shout out his dreams to people who are quickly going deaf.
Joshua Lundgren of Gun X Sword is one of the greatest mecha engineers in the world... and doesn't understand why women scream when a guy walks into the girl's bathroom.
Bleach: Orihime is a straight-A student and third best in a 300-pupil year group. She also has black-belt karate skills ( downgraded to yellow belt in the anime) and is one of the two most gifted members of the Handicraft Club. Despite her genuine intelligence, she's also a Cloud Cuckoo Lander with a very active imagination resulting in her having sometimes bizarre reactions to mundane events.
The title character of Suzumiya Haruhi is brilliant at most things she does, scoring incredibly high in all her classes and every sport she tries with minimal effort, but she utterly fails when it comes to anything that requires her to consider other people's points of view, like the movie she made. She thought it was brilliant; everyone else was... less enthused. At one point she fails to comprehend why characters always die at the climax of a story, and another time she complains that crabs didn't evolve softer shells so she could eat them easier. She also fails to realize that wearing a Playboy Bunny outfit at school would upset the teachers. And then there's her general Insane Troll Logic, such as pointing out that Tanabata wishes will take years to reach their destination due to the speed of light delay, but expecting that Orihime and Hikoboshi will be able to ignore the lightspeed barrier because they're gods.
L of Death Note is a Ditzy Genius (compared to Genius Ditz Misa Amane) as he was stated by the author of Death Note to be the most intelligent character in the series, and has enough idiosyncrasies and few enough social skills that he can barely interact. Generally hyper-competent, he tends to fail in normal conversations and interpersonal relations.
Calvin: Hey Mom, did you know that gravity in outer space works as if space was a soft, flat surface? It's true. Heavy matter, like planets, sinks into the surface, and anything passing by, like light, will "roll" toward the dip in space made by the planet. Light is actually deflected by gravity! Amazing, huh? And speaking of gravity, I dropped a pitcher of lemonade on the kitchen floor when my roller skates slipped.
Jason from Foxtrot. The kid always does his homework, and gets incredibly high marks in school, and yet will pull of incredibly dumb, and sometimes dangerous stunts, half the time just to torture his sister.
Futari Wa Pretty Cure Blue Moon has Ami-sensei. She's constantly given to speeches about hope and justice, chasing a promotion she's not sure will ever arrive despite her best efforts, and is the laughingstock of the school, but she's a Gadgeteer Genius who can build anything from a super-toaster to an elaborate trap for the local Panty Thief to an interdimensional communicator.
Brilliant physics-defying engineer Montgomery Scott, a man so brilliant he came up with trans-warp beaming. And a man so stupid, he decided to test the thing on Admiral Archer's dog, and promptly got himself Reassigned to Delta Vega.
A Beautiful Mind has John Nash, a mathematics wiz who will actually go up to a woman and say, "I don't exactly know what I am required to say in order for you to have intercourse with me. But could we assume that I said all that. I mean essentially we are talking about fluid exchange right? So could we go just straight to the sex. "
Michael Sevenson in the Knight and Rogue Series manages to be Too Dumb to Live after having gone through a university. His complete inability to even tell a partial white lie, and his tendency to act without thinking (sometimes to the extent that he doesn't even realize he's acted until after that fact), not to mention his tendency to ignore social norms, make it surprising whenever he shows that he does, in fact, posses intelligence.
Also nominated: Bellatrix LeStrange. Or possibly Helena Bonham Carter.
Leonard of Quirm is the most intelligent man on the Discworld ... and has absolutely no comprehension of human nature whatsoever. The Patrician finds conversations with him a combination of relaxing (since he doesn't have to watch what he says) and frustrating (since Leonard comes up with things like "Yes, I've invented an incredibly powerful war machine, but that's okay because no-one would ever use such a thing.")
Live Action TV
In The Big Bang Theory, Sheldon Cooper may be a brilliant theoretical physicist, but is insufferably arrogant, eternally confounded by social conventions, and almost completely ignorant of pop culture outside the sci-fi genre. However it may be that him being a genius is just an Informed Ability as he happens to think evolution is always towards superiority and has not achieved much in his field (or at least in the way of having one of theories confirmed), he even made an arithmetic mistake in one paper he wanted Stephen Hawking to read.
The Doctor in Doctor Who. He's eccentric, more than slightly manic, rarely compliments anyone's intelligence without mentioning his own, and has the ability to, without fail, sniff out and leap headfirst into any danger that his companion hasn't already stumbled into. Oh, and he's also a ridiculously competent Genius who's saved the Earth countless times, but that's nothing you'd be able to tell at first glance. Especially not if you're glancing at his sixth incarnation. Or the Fourth. Or the Tenth. Or... any version of him for that matter. But Six was especially bonkers. Eight had a definite case of Attention Deficit... Ooh, Shiny! going on, too, both in the movie and the Eighth Doctor Adventures novels:
Eighth Doctor: A meteor storm. The sky above us was dancing with lights. Purple, green, brilliant yellow. YES!
Eighth Doctor: These shoes! They fit perfectly!
Reid from Criminal Minds often qualifies, although he's working on it, in his odd Reid way. He's very good at numbers, statistics and abnormal psychology of all kinds, but no good at all, most of the time, at things like the unspoken rules of conversation and tact.
Bones. Oh, Bones. So intelligent for solving crimes and yet so clumsy with social stuff...
Because her understanding of human behaviour is more intellectual than intuitive, she forms conclusions that make perfect sense in theory, but are flawed in practice because humans themselves are flawed. It was heartbreaking when she told Booth she was going to adopt a dog that had been trained as a killer, only for Booth to tell her that the dog had been put down, because her protestations that the dog shouldn't have been blamed were right.
Kurt Hummel from Glee is very intelligent and a fluent French speaker, but prone to moments of ditziness, such as his belief boxes have four sides or his plan to feed the doves at Burt and Carole's wedding glitter.
Alex P. Keaton from Family Ties. He's a hypercompetitive straight A student and math genius who was doing his parents' taxes when he was five years old and advised his parents on mortgage rates when they bought their house. Yet in everyday activities he often proves completely incapable. His little sister frequently beats him in sports, he fails at things like building kites, cooking or fixing cars even with extensive directions.
Sherlock from, well, Sherlock. Justified as Sherlock only keeps important information in his "hard drive," which does not include tact, common sense or the idea of heliocentrism.
Philip from Kamen Rider Double has access to all the knowledge of the earth and is quite smart in his own right, able to figure out many things that Shoutarou can't. However, due to a Mind Wipe and being trapped in a factory placidly taking orders for eleven years, he doesn't know things that most people consider rudimentary. When he happens upon a subject that he finds interesting (a place, a food, a type of dance...) he will obsessively look up everything he can about it, oftentimes dressing up in zany outfits or trying nearly-suicidal activities just because he can. Also, especially towards the beginning of the series, he has No Social Skills.
Recurring Character Professor Pepperwinkle in The Adventures of Superman. Example: he brought his latest invention to show Perry White. It was an (untested) time machine. Perry didn't have time to look at it because he was busy with an Important Story involving a gangster who was about to turn himself in to The Authorities, and who was in his office at the time; so the Professor turned his time machine on and sent them all (himself, the gangster, Perry, Lois, Jimmy, and Clark) to prehistoric times. Success!
...then he realized he didn't bring enough fuel to make a return trip.
The Solomon's from 3rd Rock from the Sun are all advanced aliens who could make fools out of even the smartest human beings, yet, because they have never dealt with emotions before, they come off as utterly crazy and weird to everyone around them. Dick, in particular, is a pompous manchild who can't even teach a young girl to play hopscotch without turning it into an argument.
Leo in Leonardo is possibly the smartest person in the world, but is incredibly easy to manipulate due to automatically believing in people, much to the despair of his Street Smart friends.
Commander Data of Star Trek: The Next Generation represents the pinnacle achievement of robotics, and is very likely the most intelligent officer in all of Starfleet. He is also routinely stymied by basic human interaction and language.
Chief O'Brien: ...We'll be burning the midnight oil on this one.
Data: That would be inadvisable...If you ignite a petroleum product on this ship at zero-hundred hours, you will activate the fire suppression system, which would seal off this entire compartment.
O'Brien: That was just an expression.
Data: Expression of what?
Data: It appears we will be required to ignite the midnight petroleum, sir.
Moss from The IT Crowd is an extremely book smart technological genius, with No Social Skills and a rather absurd lack of common sense. When a fire broke out in his office, for instance, his reaction was to send a carefully worded E mail to the fire department, then sit there and wait for them to respond.
Dr. Shaw in Dino Attack RPG is a very intelligent and capable surgeon and even shows some talent as a counselor. On the other hand she's extremely unstable psychologically and spends a lot of time praying to a being she only half-believes in.
Arguably, Otacon in his first appearance in Metal Gear Solid. The man's brilliant with technology, but social interaction and judging the motives of other people are not his strong points.
Dr. Ivo "Eggman" Robotnik is a genius who has an IQ of 300, capable of singlehandedly building entire fleets of airships and space stations as well as armies of killer robots. However, he usually dresses his machines in garish colors, tends to throw temper tantrums whenever he's on the losing side and has a tendency to paste his face on everything.
In addition to almost always including an incredibly obvious weak point in all of his robots (which is not unusual or unexpected for video game villains), special mention must be made of the Egg Dealer - a slot machine robot that had a random chance of attacking himself or even giving Shadow immediate access to his ultimate technique. Got to wonder what he was thinking when he programmed those particular features in.
Rita Mordio in Tales Of Vesperia. A prodigy mage at the age of fifteen, but possessing absolutely no social skills. She's rude, direct, violently assaults those who bother her and cares nothing for other people. This can be chalked up to her young age and that she was raised (as an orphan) in a university town filled with self-absorbed scientists.
Pascal in Tales Of Graces is a lot more sociable than Rita, but she's so energetic that she's nigh-incomprehensible when she gets excited, and she can't read social situations: she doesn't so much break the tension in a room as much as crash through it head-first. She also has no concept of personal space.
Harold from Tales Of Destiny 2 is an eccentric, unpredictable, childish and sometimes downright psychotic young lady; yet, she is the genius behind the creation of the Swordians, and a magic expert. The writers of Vesperia seem to have taken a bit of her when making Rita; theory supported by a Harold mask usable by her (as an Attachment) and her Miska Doctoral Degree costume title, very reminiscent of Harold's garment.
Luigi may qualify for this. When he's turned evil, he manages to build a giant killer robot that takes over two hundred hits with an alien space laser to go down in a couple of hours, owns his own airship (that he's implied to have built himself), is a master at card games, is explicitly stated to be the brains of the brothers, when Mario has a job which already requires someone to be very smart, and, yet, acts extremely childish when he's on adventures. On the other hand, it might all be an act to make himself seem less capable.
Merrill of Dragon Age II is a brilliant mage and historian well noted for her talents. She's also extremely naive, very socially awkward, and has little in the way a common sense (for some time she considered muggings to be some form of greeting). This is both played for laughs and treated seriously.
Medic. Oh, Medic. He may look like he has it fairly together on the field, but look a little closer and you'llfindout otherwise. For example, he apparently believes that the best way to talk to a patient during surgery (the patient being his Heterosexual Life Partner, no less) is to regale him with the story of how he lost his medical license (it involved removing the entire skeleton from a man who lived to tell the tale). It's fairly obvious that the man is both socially inept and generally round the twist.
Australium poisoning has this and Testosterone Poisoning as its main effects. Radigan Conagher (Engie's grandfather), after working with australium for years, became intelligent enough to make a fully functional prosthetic limb, and dumb enough to saw his hand off to be able to use it, and Australia, the main source of this metal, is the most advanced country in the entire world, yet they elect their king by kangaroo boxing.
The Sophon's of Endless Space, they're just as likely to run away screaming from a planetary anomaly or study it despite being irradiated.
Yukiko Amagi of Persona 4 has a bit of this. Intelligence-wise, she's very smart, as her best friend Chie remarks that Yukiko always ranks near the top of the school charts when it comes to exams, and she's also often the one to make important deductions about the murder case the group is working on (up until Naoto joins the group, anyway). On the other hand, she's capable of being very air-headed too, often displaying a remarkable ability to miss the entire point of a conversation (for instance, not knowing that a guy asking her to "go somewhere" with him was asking her out on a date, or thinking Yosuke is making a dirty remark when he suggests Yukiko give him some "private lessons" during a conversation about studying), and occasionally spaces out completely when others are talking to her.
The writers of Tsukihime were apparently quite aware that the average reader was likely to dismiss Arcueid as an idiot. In truth, she's actually very intelligent and knowledgeable, she's just never had to pick up the skills that Shiki takes for granted. Which she points out. Plus, she's still a ditz.
Kotomi in Clannad regularly tops or at least places in the top ten of national grade/test averages and studies incredibly advanced physics in her spare time. She also doesn't seem to see much problem with using scissors on any science book that mentions her parents or their work and reacts... very... slowly... to things, plus manysimilartraits that become more obvious with time. Watching her 'tsukkomi lessons' with Kyou and Nagisa can be both facepalm inducing and hilarious.
Palma from Mushroom Go. She's an immature, loud mouthed teenager with a short attention span and poor grammar. She may also be the smartest Pianta in the world, and has spent most of her life lost in books.
Brandi of Wapsi Square is beyond brilliant when it comes to planning, to the point where any unusual coincidence that worked to the favor of the protagonists can be attributed to her manipulations. She also fails to realize that flies drown when she flushes them down the toilet (she couldn't just shoo them outside because it was winter in Minnesota, and they would quickly freeze to death).
Link is this in Nintendo Acres: Competent at being the hero of The Legend of Zelda games, has an incredible knowledge of literature, and can't open a door without using a bomb or boomerang.
Brian Fitzpatrick in Rhapsodies is a gifted polymath, economist and accountant who can pretty much make numbers do what he wants. He is however impulsive with an extremely abrasive personality and is so socially clueless most people think he's an idiot.
So and So from Teen Girl Squad. She gets straight A's and prides herself on being an academic overachiever, yet is utterly devoid of anything resembling common sense.
Delta Spike from Whateley Universe is a prime example: a Gadgeteer Genius who is especially well known on campus for the catastrophic explosions her devises cause when they inevitably fail, she tends to assume that everything will work perfectly and doesn't consider consequences. As one character put it, "She knows A plus B will equal C, but she fails to consider that there's a D, E, and F."
Gordon Freeman as depicted in Freeman's Mind is a narcissistic, bumbling,lazy, and overall bizarre human being who, among other things, built pillow forts out of plaster when he was a kid, got mad at zombies for not answering his questions, thought his mailman was spying on him, and stayed up for three days straight when he thought his house was being invaded by frog people. Some of his interactions with the scientists also show him to pretty inept and overall dickish socially, and he's implied to go on drinking binges regularly as well as get high on whatever's available. However, his inner monologues show that, despite all of this, he's quite competent in combat, is actually very knowledgeable about theoretical physics, speaks at least five languages (Haitian Creole, Spanish, German, Hindi and English), and has proven consistently more competent both physically and mentally than anyone else in Black Mesa.
Shadowcat from X-Men: Evolution also qualifies. She's portrayed as a very intelligent student, but could be easily excited over "typical" teenage topics (driver's license) and early on had the tendency to use the word "like" so much in a single sentence, Shaggy Rogers would take notice.
Bessie Higgenbottom can do anything that will earn her a badge - including creating cold fusion or building a working robot with limited AI - but that doesn't make her any less of a ditz.
Twilight Sparkle from My Little Pony: Friendship Is Magic. A unicorn so smart, she knows most of the constellations by heart and was able to finish a race fifth place, ahead of a few dozen other, more athletic horses, simply by using her book smarts. She’s also a filly so silly, she needs to consult a reference book on something as simple as a sleepover. She even reads the book to find out what to do when a tree falls into her house, instead of just getting up and helping move it.
The third season of Generator Rex gives us Cesar Salazar, the formerly absent older brother of the eponymous Rex. Among his many achievements are: Working on the nanite project alongside his and Rex's parents, building a mobile lab capable of traveling at relativistic speeds, sauntering into Providence and casually neutralizing everyone between him and his kid brother, and creating the insane AI, ZAG-RS, which has nearly succeeded several times in destroying all life on Earth. When not performing feats of scientific awesomeness, however, it's clear that, against all logic, Rex, not known for being the smoothest guy around, got all the social skills in the family.
Jack Spicer from Xiaolin Showdown, to the extent that the other characters tend to forget that he really is a genius.
Roba from The Problem Solverz. He's the smartest member of the team, is knowledgeable about many subjects, and has built his own radar scanner. However, he's also very socially inept and childish.
Heinz Doofenshmirtz from Phineas and Ferb. He's a genius when it comes to science, technology, math, and engineering, but he lacks just about everything else such as social skills and common sense. He puts self-destruct buttons in a convient location on just about every invention he builds, and he's never even recognized his nemesis, Perry the Platypus, in even the most transparent of disguises, and tends to assume Perry is just an ordinary platypus if he's not wearing his trademark fedora. To put all this into perspective, take his scheme in the episode "Fireside Girl Jamboree": He was smart enough to build a machine that is capable of transforming metal into broccoli (A mineral into living matter), but decided to make it out of metal.
Child Prodigy Dexter from Dexter's Laboratory definitely fits the bill. He's able to time-travel, create robots and build portals to other dimensions, but is utterly incapable of taking care of himself for a single day when his mom is sick. He doesn't know how to cook (having never heard of "flour"), and as for cleaning...
Plankton in Sponge Bob Square Pants, despite being an evil genius he tend to over look simple things to which his computer wife Karen has to point out. Like the alphabet ends with Z, and he doesn't remember how to blink.
Professor MacKrill from Help Im A Fish he's a highly intelligent scientist who created a potion that can be used to turn other living beings into fish and vice versa, he easily forgets things and often makes up songs to help him remember.
Po from Kung Fu Panda: Capable of recreating Forbidden Techniques from inference or entirely on his own. Competent strategist. Encyclopedic knowledge of kung fu history, philosophy and lore. Achieved enlightenment in three days. Supreme Chef. Skilled woodworker. Childish, clumsy, easily excited and dangerously lacking in common sense.
There is an old joke about a professor who, at a university formal dinner, helps himself to some peas using his hand. When he realises what he's done, he excuses himself with "I'm sorry, I thought they were beans."
Likewise, there's the old fable about an astronomer who is so busy studying the stars that he fails to pay attention to what's around him and falls down a well.
Some Dutch used to say: "Krankzinnigheid ligt dicht bij begaafdheid". This means as much as: "Crazyness is close to genius."
Their ethnic cousins, the English, have a similar chestnut: "There's a fine line between insanity and genius."