Many cultures in Real Life hold that it's not polite to point. In a lot of western cultures, this stems from a superstition that finger-pointing was how witches would jinx people. Of course, the fact that aggressively thrusting a body part at someone could possibly serve to annoy them probably has something to do with it.
In real-world Japanese society, personal space is at a premium and thus of the utmost importance. One is careful to avoid even the appearance of proximity, and one keeps one's hands at one's sides. Thus, in animeland, it is a gesture of extreme emotional importance to physically point at someone. This should be saved for only the most appropriate of times. If breaking out the pointer finger, it is traditional to be especially dramatic about it. A stylized "swirly" fingerprint symbol on the fingertip is optional.
However, most characters who Give the Pointer Finger aren't really doing it in an appropriate situation, which can signify either immaturity or status as a Large Ham.
Often used to denote a new rival or Unknown Rival, although more often than not it is simply the trademark of a hammy character.
Generally unrelated to giving someone the Finger Poke of Doom, nor is it really related to giving someone another finger.
Compare Sword Pointing, which is similar, but done with melee weapons.
Protagonist Judai does this constantly in Yu-Gi-Oh! GX, always accompanied by his Catch Phrase: "Gotcha! That was a fun duel!"
Haruhi Suzumiya does this a few times in a non-homage context, because she's loud, brash, and doesn't care about personal space.
Tsuruya has a tendency of doing this when she laughs.
Used by Ichigo in Bleach when rescuing Rukia Kuchiki. When she started to object that he should save himself, he points at her and tells her that her opinions have been rejected.
Also done when Ichigo and Ganju meet unexpectedly in Kuukaku Shiba's house. Hilarity Ensues.
Used by Amelia in The Slayers when she is introducing Zelgadis and Gourry to her father, and trying to rally them to return to the fight with Zanafar. She also poses and points dramatically when they finally do confront Zanafar again. None of them pay attention to what she is saying and all of them comment on how impolite she is being, including Zanafar.
Chii from Chobits, frequently does this — mostly while repeating whatever tidbit of information she just learned. Chii, at least, has the excuse of being an android who doesn't know proper etiquette.
And she's too cute for anyone to get mad at her for long.
Hikaru from Himechan No Ribon does this extremely closely to Hime-chan's face when declaring war over Daichi.
Not to mention Monta, who even has the swirl on his fingertips, even when he's wearing gloves.
Kabuto Kouji from the Mazinger trilogy -Mazinger Z, Great Mazinger and UFO Robo Grendizer- loves doing this. A lot. Especially in Mazinkaiser. If there aren't a bare minimum of two instances of this in every episode, then you're watching the wrong anime. Usually accompanied by an In the Name of the Moon speech which will generally go something like "This Mazinkaiser will be your opponent!" The supporting characters tend to join in.
Honorable mention goes to Tetsuya's Great Mazinger, whose Thunder Break attack produces and absorbs a lightning strike, then fires it out the pointer finger.
Duke Fleed, the main character of the third series, sometimes fell into this, too.
Allison from Allison and Lillia does this a fair bit; as a Genki Girl who clearly wears the pants (flight suit, actually) in her relationship with childhood friend Wilhelm, it's practically a given.
Naruto does this a lot. Being the brash, immature loud-mouth that he is, it's not surprising that he doesn't have manners, either. In the original before Shippuden, he usually pointed when accusing someone (usually a teacher or Sasuke) of undermining him or doing something behind his back.
Sakura does this to Hinata in one Naruto Shippuden omake when suggesting that the chocolates she's making for Valentine's Day need an extra touch (unfortunately, the nutritional supplements she adds end up making them taste disgusting) to appeal to the recipient.
Itachi has a more reserved, but far more weaponized version of this. When he points his finger at you, you've probably just been caught in a genjutsu.
Though he has to stop gesturing altogether after he gets super-strong.
Also, John does it to As'taris just after getting the Kansael. While jabbing his finger at the elf, he accidentally squirts him; and when he looks at his own finger, he squirts himself.
Turnabout Storm: Phoenix Wright makes liberal use of them, they being his signature gesture after all. This time it's more jarring than usual, given the general lack of any fingers to speak of on the inhabitants of Equestria.
Trixie: Don't point that ugly sausage of a finger at Trixie!
In the film Ghost Rider, the titular character often does this, even when he really oughta know better (it's probably not wise to point and say "YOU!" to Satan himself). He once points threateningly to a young child while transformed, and then says "innocent!" and walks away. Said child passes out from the fear.
In Practical Magic, Sally lowers her daughter's hand to stop her from pointing—but her daughter does it anyway. It actually was a Magical Gesture, but it's cute given that the worst the girl can think of is "I hope you get chicken pox".
The 1978 version of Invasion of the Body Snatchers has the alien replicas pointing and screaming at normal humans in order to mark them for the other aliens.
At the end of the first Harry Potter book, Ginny excitedly points at Harry after he arrives back at King's Cross and Mrs. Weasley tells her that it's rude.
In ''For The Emperor'', Inquisitor Vaile confronts the treacherous Governor Grice, "pointing an accusing finger at him like a schola tutor admonishing an unsatisfactory student." Of course, she was wearing a digital needler that shoots poisoned darts.
A Song of Ice and Fire. King Joffrey points accusingly at Tyrion Lannister as he's dying, making his mother the Queen Regent believe that Tyrion poisoned him. After his arrest further "proof" of Tyrion's guilt is quickly assembled given the long-held antagonism between Tyrion and Joffrey, though Tyrion is in fact innocent of the crime.
In the DVD commentary for the first season of The Mighty Boosh, Noel Fielding describes Matt Berry as a 'pointing actor', as he tends to use this trope extensively in the roles he plays, particularly in Boosh and The IT Crowd.
A rather tongue-in-cheek example of this is in a Power Rangers in Space episode where Zhane is disguised as the Silver Psycho Ranger. When the Blue Psycho Ranger grows aggravated with Zhane's actions, he points at him intimidatingly. This provokes Zhane to respond in a very cavalier manner: "Ah, ah, ah. It's not nice to point."
"You raped her. You killed her children. Say her name: Elia Martell. Who gave you the order? (points at Lord Tywin) WHO GAVE YOU THE ORDER?!"
The Wicked Witch of the West periodically does this to the player in Jersey Jack Pinball's The Wizard of Oz.
Western example: uber-star Hulk Hogan did this all the time when he made his big comeback. Inevitably, after the fingerpoint (punctuated by the crowd yelling "YOU!"), the match would be won within a minute.
Phoenix Wright's signature move from the Ace Attorney games whenever making an exclamation ("OBJECTION!") is to pound the stand and point. Heck, everyone in those games do. It's their job. The game logos in all regions even feature the title character's silhouette striking a dramatic pointing pose. This is recognized enough that Haruhi paid homage to it during the "Murder Mystery" episode. There is massive Lampshade Hanging over the constant forceful pointing and yelling.
Some anime characters would be concerned about High-Pressure Blood spray if they cut their fingers. In Case 1-5, Phoenix is concerned about not being able to point at people.
Defense lawyer Calisto Yew strikes this pose near the end of one of your rebuttals in Investigations, with the added twist that she's holding a gun.
Miles Edgeworth lampshades this at one point in case two with the line "Don't fall asleep while I point my finger at you!"
Daryan Crescend does it in the rude fashion when making fun of Apollo Justice.
Gyakuten Kenji 2's President Ō uses a distinctive underhanded variation, which makes it look like he's Flipping the Bird at Edgeworth (which, in context, is pretty much what he's doing) even though he's using his pointer finger like everyone else.
Actually subverted in Dual Destinies! Prosecutor Simon Blackquill tries to...but the chain of his shackles prevents him from being able to, resulting in a point towards his desk rather than the defense. When he breaks said shackles, he doesn't even bother with this, just launches a Razor Wind at the defense and/or the witness.
This little move is a trademark of the Professor Layton games. When solving puzzles, Layton, his apprentice Luke, his ward Flora, his other apprentice Emmy, and Luke from the future all do this to the player, accompanied by one of their assorted catchphrases. Layton also does this during the Summation Gatherings when he accuses the guilty party in each game.
Even Phoenix's introduction in the trailer starts with him doing the desk-slam and point.
During certain points of the witch trials, they turn and point in unison. Taken to humorous levels in "The Golden Court" where they do the same thing despite the fact that Layton is currently a gold statue. Luke and Espella turn the statue (which is pointing, of course) to match Phoenix's movements. At the end of the game Layton even prevents two characters from falling to their deaths by pointing at them until they stop in mid-air. The invisible crane may have had something to do with it, but still...
Battler from Umineko no Naku Koro ni gives Beatrice this finger a lot, usually as a challenge, similar to the Phoenix Wright example above. The emotion behind it is actually fairly appropriate, though, given the context.
Well, it also doesn't help that Battler looks like Phoenix.
All because of a typo in the original Tsukihime where normally benevolent Meido Hisui says "You are going to be the criminal" as a prediction instead of an accusation. This lead to a Memetic Mutation where a darker version of Hisui called the "Brainwasher Detective" would point at people and spin her finger to hypnotize them into being criminals so she could arrest them. This occasionally shows up in Mythology Gags such as her attacks in Melty Blood or Battle Moon Wars.
In the CD-i game, Hotel Mario, Luigi does this near the end of the opening cutscene:
Metal Gear Solid 4's Liquid Ocelot loves to do this. In fact, expect a dramatic gesture or pose every 50 seconds at the least in any scene he features in, 11 in the space of one scene. But then, Liquid Ocelot is a Large Ham, who loves to Chew the Scenery.
When Liquid Ocelot did this in Act 3, it resulted in several choppers crashing, some gunfire from his troops, and (via a hand gesture resembling a gunshot to the head) several seizures. It's strongly implied that these were caused by Liquid Ocelot shutting down their dependence on the SOP system.
Mass Effect 2: If Legion is Shepard's second squadmate during Tali's loyalty mission, a few additional seconds of cutscene-footage are added where Admiral Koris does this to Legion at the very beginning of the trial.
A couple character portraits in Infinite Space point at the player, including Commander Vladykin of Elgava and Coyle, a Lugovalian administrator.
In The Legend of Zelda: Skyward Sword, Groose shoves his finger right in front of Link's face while bragging about how he is going to win the upcoming race now that Link can't get to his Loftwing. When Zelda comes across Link and Groose seconds later, she does the same gesture to Groose while warning him to leave Link alone.
For the Brits, Lord Kitchener, who wants YOU to join your country's army.
Averting this trope is considered common speech etiquette. Bill Clinton is most famous for averting it, using the common technique of pointing with his thumb pressed against the first joint of his index finger, which became a hallmark of his impersonators.
At Disney Theme Parks, "cast members" are trained to gesture with their whole hands, mostly for the benefit of international guests whose cultures follow variations of this trope.
In the US Military, a common subversion is to avoid pointing at someone with your finger, particularly if you are correcting them. Instead, it is common to point with your entire hand, in a gesture known as "Knife Hands". Use a "karate chop" motion or angle the hand more aggressively for added emphasis.
A lot of people criticized Arizona governor Jan Brewer for pointing her finger in President Obama's face during a meeting. A lot of people saw some Unfortunate Implications, saying that if she were a man or if Obama was white, she never would have had the audacity to do it in the first place.
In Northern Ireland, the act of emphatically jabbing a pointed finger at the person you are talking to so as to reinforce the point is culturally acceptable and not seen as anything out of the ordinary. British soldiers had to be explicitly told this was not meant to be an aggressive or disrespectful thing - just normal everyday Norn body language. Witness the (now deceased) Reverend Ian Paisely in full oratorical flow.