So, Mr. Exposition's here and he's about to give a quick introduction to the magic system, outline his plan to defeat the Big Bad and explain where those nasty beasties hail from. Unfortunately, it's hard to explain this to the audience. It may be too complex (or dull) to explain in just words, or perhaps the producers don't want to be accused of resorting to Mr. Exposition talking for 5 minutes. Either way there's an easy solution: draw them a diagram. A diagram for the purpose of exposition.
Note that this only applies if the diagram is there for the benefit of the audience. Whether or not the cast themselves can see it is incidental (so a diagram which the audience can't see or isn't part of any exposition doesn't count). Variations can include examples where:
The diagram really exists and was created normally (e.g. by drawing it).
In-universe drawn diagrams are often used in to explain concepts (such as hollows in the very first episode/issue). If Rukia's around (and she will be) she'll draw the diagrams herself, using poor drawings of rabbits. Much to Ichigo's annoyance.
Anime episode 22: Ganju Shiba uses the visible only to the audience sort (with Japanese writing) while introducing himself.
Anime episode 23: Orihime Inoue, Chad and Uryu use Type 3 diagrams when discussing what Yoruichi's friend Kūkaku Shiba looks like. Then Orihime draws a Type 1 diagram in the dirt to combine their ideas together.
Anime episode 33: Don Kanonji a chart appears to detail the strategy of the Karakura Superheroes against a giant Hollow.
Anime episode 146: Nel explaination of the nature of numbered Arrancar is accompanied with diagrams for the audience.
Anime episode 193: Szayelaporro Grantz uses a slide show to demonstrate the difference between the protagonists' and their clones to the audience.
Anime episode 197: During the fight between Espada #7 (Zommari Rureaux) and Captain Byakuya Kuchiki, Rureaux gives Captain Kuchiki a lecture about control and the nature of his special eye power.
Soul Eater had these on occasion, such as when Death the Kid explained why he preferred the number 8 to 7 (because it's symmetrical, for those who don't know), and more seriously (or as seriously as you can get with little chibis running into each other) by Dr. Stein when he outlines his plan to prevent the Kishin from being released.
In Naruto, the explanations of how chakra is generated and manipulated are often accompanied by diagrams
Audience only diagrams are also used when discussing strategies.
In One Piece, inset panels explaining their Devil's Fruit abilities accompanied the reintroduction of Buggy the Clown, Mr. 2 Bon Clay and Crocodile.
In another example, Eva explains how Negi uses her resort to pack two days into one.
Heck, this is so common in Negima that even the Bad Guys use it to explain things.
In Liar Game, when Akiyama/Yokoya begins drawing one of these, expect the other side to go Oh Crap.
In Change 123, when the Insufferable Genius Kannami explains certain aspects of Motoko's Multiple Personalities (and once when explaining Hibiki her mother's family history), the audience only diagram is sometimes used, in form of a mixture of schematic diagrams and portraits of various characters/personalities, and once a diagram drawn by Kannami himself on a sheet of paper.
Slayers Next uses an audience only diagram after the filler arc. At the beginning of the episode that returns to the serious plot, Lina recaps the situation with Gaav while a helpful diagram displays the Mazoku lords and their relations to each other.
Unexplained and audience only examples tend to pop up all over the place in Ranma ½. They appear out of nowhere, and are gone in the next scene. Sometimes the diagram obviously doesn't actually exist, but the characters turn around and react to it anyway.
Episode 4. Ryuuka uses a series of audience only examples to explain the rivalry between the Jihyou and Hanaukyo families.
Episode 7. While in a hot springs house Taro's 3 personal maids hold up signs they pull from nowhere to describe the nature and beneficial effects of the hot springs waters.
Episode 10. An audience only example when Mariel lists Taro's daily activities and they appear (as if being typed) in the air over Taro's head, gradually squashing his head down.
La Verite episode 5. While the protagonists are at Comiket a message in Japanese appears over Mariel's head.
La Verite episode 7. During an underground expedition two of Taro's personal maids hold up signs (which they pulled out of nowhere) with Japanese writing indicating they give up and don't want to go any further.
Episode 3 "Marquis de Carabas". While Naota is talking to Haruko about his father, an audience only sign appears with his father's face on it.
Episode 5 "Brittle Bullet". A blackboard appears while Haruko is wearing the Elvis Presley costume.
In The World God Only Knows, Katsuragi Keima uses several diagrams to explain his "Conquest" strategies. Probably the most recognizable of these is the diagram explaining his flower metaphor which appears in the Ayumi and Chihiro arcs. According to the diagram encounters with the heroine function to water the flower of love. "It needs lots of water to bloom!"
In Kurosagi, whenever Kurosaki describes a swindling job a diagram appears to help the reader follow his words and get the flow.
Saijou No Meii: Imaginary diagrams are used in almost every chapter in the series to explain medical procedures, politics, and terminology.
This was often a feature of JLA/JSA team-ups in DC Comics during the '60s and '70s. Since the teams existed on Earth-1 and Earth-2, the stories invariably included a page or so explaining the Alternate Universe concept for those who are unfamiliar with it. Often, this would involve Dr. Fate or some other character and a visual literally showing the multiple versions of Earth with comparative illustrations of, e.g., the Flash of Earth-1 and the Flash of Earth-2. The issue of whether the illustrations and diagrams actually exist In-Universe or not is usually kinda fudged.
Parodied in Family Guy: "...and now we have a black board in our living room for some reason."
In Memories of Nobody (one of the Bleach Movies), Kisuke Urahara uses a flipchart to explain to Ichigo what's going on.
Beetlejuice. Charles Deetz uses an in-universe diagram to show what his "Museum of the Paranormal" will look like.
Disney's Aladdin. The Genie conjurs up a blackboard that's hanging in the air to advise Aladdin about how to handle Princess Jasmine. It says "Tell her the..." [flips over] TRUTH!"
Gru in Despicable Me explains his plan to steal the moon. The girls have inserted a drawing of him on the toilet into it.
The Narrator in The Hallelujah Trail would use animated maps showing the movements of the various groups involved, until the situation got so confused that even the maps didn't help.
Jay and Silent Bob use diagrams in Mallrats to describe how they'll sabotage the dating game show.
Live Action Television
Possible aversion or subversion: The 2000's Battlestar Galactica had a scene where Crash, leading a group of mixed soldiers and civilians against the Cylons, made diagrams for his troops out of sticks, rocks and leaves. The others really, really tried to take him seriously, but as most of you can imagine, it was really, really hard. The scene illustrated how Crash, despite being the highest ranking officer of the group, had very little actual ground combat experience.
Stargate SG-1 does this A LOT, although usually Mr. Exposition's drawn "diagram" is actually a flashy computer animation s/he threw together in five minutes. Examples include: Jackson in the first movie explaining the 7-symbol address system; Carter explaining how many Tollan ion guns they'd need to defend Earth; Zelenka explaining why Atlantis' self-destruct would leave chunks large enough for the Wraith to salvage Ancient technology; Dr. Lee explaining how to tell Atlantis they have a bomb in the city without directly dialing their Stargate; Rush explaining that Destiny will leave the galaxy soon, but that it's still possible to send a team backwards through the gates to rescue the stranded main characters.
Parodied in the Father Ted episode "Speed 3". When Ted has a Eureka Moment, he starts furiously scribbling on a blackboard while narrating his train of thought. After this, we discover that the "diagram" consists entirely of the text "WE PUT THE BRICK ON THE ACCELERATOR".
On How I Met Your Mother, Barney explains the correlation between a woman's attractiveness and insanity with the Hot/Crazy Scale, which he draws on the air and is visible only to the audience.
The trope is inverted in another episode, where Marshall has a number of charts and graphs that are real and of a clear origin, but doesn't use them to explain anything important. He just really likes charts and graphs.
Scrapheap Challenge used these on occasion when explaining the physics and/or engineering principles being used for this week's project.
NUMB3RS used these Once an Episode, in the form of an Imagine Spot of whatever analogy Charlie (or occasionally Larry or Amita) was using for his math, overlaid with some relevant (or at least relevant-looking) equations and graphs and stuff.
Leko uses an in-universe diagram in episode 30 of Comic Shorts: Spriteoverse to explain the multiple courses of action the characters can take to deal with their current situation, complete with cutting off Tailii's attempt to ask where the diagram came from.
Dr. Drakken does it a few times in Kim Possible, once displaying the plan to derail a train and steal its cargo with a train set; one when he just happens to be carrying a diagram showing he and Shego breaking into a lab stealing Ray X. Another time, having captured Kim, he pulls up a flipchart to explain the particularly elaborate belt-and-braces approach of the Death Trap he's going to dispose of her with.
One time in Phineas and Ferb, Dr. Doofensmirch catches Perry in a trap and begins to explain his take over the tri-state area plan. The last slide is a picture of himself explaining the plan to Perry.
Even better was the time he did his evil monologue in pop-up book form, which again ends with him explaining the plan to Perry.