In the first season finale, Pakku tells one of his students that in a few years he might be able to defeat a sea sponge. The creators later admitted in a commentary that it was a reference/friendly poke towards the fellow Nicktoon, one of its main competitors. The second season also contained a shout-out to its cancelled, fellow Periphery Demographic Nicktoon Invader Zim, for whom Avatar co-creator Bryan Konietzko had been an art director, as well as numerous shout-outs to its fandom, especially to the cosplayers and fangirls.
"Nightmares and Daydreams" has Aang imagining Momo as a samurai, who looks very similar to Miyamoto Usagi..
When Toph tells Aang, "You da man, Twinkletoes!", she flashes the Buddy Christ sign at him. It's a little difficult to make out, since she's aiming right at the camera and not slightly to the side, but it's there; she's pointing with the right hand and holding a thumbs-up close to her body with her left.
The Sandbenders, bandage-wrapped desert nomads, look like the Tusken Raiders of Tatooine. Possibly a double shoutout, because not only did the Sandbenders' appearance call to mind the Tusken Raiders (sand people, sandbenders... huh.), but the way they were shown in their first named appearance (where the archaeologist shoos them away from Appa outside the ice spring fruit bar) is very similar to the treatment of Jawas.
The Boulder's name is possibly a reference to Hulk Hogan, who first wrestled under the name Terry Boulder, though it is a more direct reference to The Rock.
Fire Nation Man is also a clear reference to Nikolai Volkoff with the use of his Fire Nation flag and his obnoxious national pride.
The creators stated that they were also fans of Cowboy Bebop, and decided to have the forest where Aang and Ozai fight named after the series currency, Wulongs. Then there's Sokka's Cactus Tango and a character named Jet (who also resembles Bebop's protagonist Spike).
Wan Shi Tong, the Knowledge Spirit who looks like a giant barn owl. When he walks with his back turned, he looks almost exactly like the character No-Face from Spirited Away.
The two "Siege At The North Pole" episodes are chock full of shout-outs to another animated show about a plot to kill a spirit, Hayao Miyazaki's masterpiece Princess Mononoke.
A scene from "The Blue Spirit", the one where Zuko is knocked out by the impact of an arrow in his (masked) face is nearly identical to one in Princess Mononokewhere San is knocked out by a bullet.
Also from Miyazaki, Appa's six legs are reminiscent of the Catbus from My Neighbor Totoro, being somewhat based off it.
A lot of the spirits - like the swamp monster and Koh - are similar to the designs of spirits from Ghibli films, since the creators are big fans of ones like Princess Mononoke and Spirited Away.
When Aang successfully defends the Northern Water Tribe from a Fire Nation siege by having the Ocean Spirit merges with him, becoming a giant, glowing aqueous creature that is an obvious shoutout to the Forest Spirit's night form in Princess Mononoke. The concept of a human trying to kill a nature spirit also comes from the movie.
Katara parting the waters in "The Serpent's Pass" doubles as a shout out to both The Bible and The Prince of Egypt. The latter only because of the serpent's silhouette showing the waters in the same manner that a shark's did in the movie.
The gate to the Serpent's Pass tells people to "abandon hope" as do the gates of Hell in Dante's Inferno.
Zuko's story is an awful lot like Anakin's in reverse. He's almost an Anakin Expy, in fact.
The series finale features two duels running in parallel. The first is a battle for political control fought in the seat of political power, the other is a battle to decide the fate of the world in a land of fiery Hell. This almost perfectly echoes Revenge of the Sith with the exception that the good guys eventually win both duels
Aang being trained by the Guru ends with him seeing a vision of his friends in danger whilst meditating and prematurely leaves before his training is complete to help them. The parallels with The Empire Strikes Back should be obvious. And in both cases, the result is not good.
The Play version of Ozai's Angels do a pose pretty much identical to the opening from Charlie's Angels after Azula kills Aang with lightning
In the episode "Jet", the fight scenes in the trees were inspired by Naruto, where many of the series' fights take place in gigantic trees.
When Sokka is throwing around names for the team at the end of "The Drill", he suggests "Fearsome Foursome", which was the nickname of the legendary defensive lines of the NFL's Los Angeles Rams in the 60's and 70's.
Toph bends the space rock Sokka gave her into what looks like the old Nickelodeon logo.
The Dancing Dragon form in "The Firebending Masters" is nearly identical to the last step of the fusion dance.
Given their Mongolian names (Mongke being named after a grandson of Genghis Khan), the comment about Rough Rhinos being "a very capable singing group" appears to reference Dschinghis Khan. For that matter, the name of their team is likely a reference to the historical US cavalry unit called the Rough Riders.
Aang's armor in "Sokka's Master" calls up a lot of various anime and fighting games, the most notable of which is Nightmare's arm piece and helmet from Soul Calibur. Also worth mentioning from that sequence: Tenchi's "Wind Sword".
Combustion man: Strange, stoic killer + guy with tattoos that make things explodenate + guy with mechanical limbs because his original ones were lost in an accident. That sounds a lot like Fullmetal Alchemist.
In Ba Sing Se, Zuko disguises himself with a mask that's actually Pazuzu, as seen in The Exorcist. However, that film actually uses a Japanese hannya Noh mask, which is in fact Asian. Thus, it's the Fantasy Counterpart Culture's equivalent foreign face.
In "Nightmares and Daydreams" Aang has a dream in which Momo and Appa are dressed up as samurai and fighting each other. Samurai!Momo duplicates the distinctive gestures made by the hero of Kurosawa's Yojimbo.
Jet bears a striking resemblance to Spike with his hairstyle, laid-back attitude and bent grass stem/cigarette in his mouth. Spike's partner is named Jet too.
The scene in "The Blue Spirit" where Aang asks Zuko if they could have been friends if they met earlier in life is based on a similar scene in the episode "Waltz for Venus" of Bebop.
The huge rock forest where Aang fights Ozai is called "Wulong Forest" after the currency in Cowboy Bebop.
The scene where Zuko, after being hit by Azula's lightning, lies on the ground twitching as lightning continues arcing across his body is highly reminiscent of Return of the Jedi.
The scene in the Season 3 episode "The Firebending Masters" where an idol pops up out of the ground and Aang says, "I'm just very suspicious of gems sitting on giant gold pedestals" is a shout-out to Indiana Jones. In the same episode, Prince Zuko wall-runs over a pit of spikes.
During their Agni Kai in the Season 1 episode "The Storm", Zuko tells his father, voiced by Luke Skywalker himself, "I will not fight you" in the exact same intonation Luke does.
The creators are also huge fans of Gainax productions, particularly Neon Genesis Evangelion and FLCL. The art book for the show lists Hei Bai's design as being based on the Angels whereas the long-limbed and hunched look of Bumi's body was inspired by the Evas, themselves. The book also mentions that the image they drew of Aang with glowing eyes as part of their initial pitch to Nickelodeon was inspired by Naota.
Aang's pushups in "The Warriors of Kyoshi", going from using both arms, to one arm, to simply blowing air, are likely a shout-out to Kung Pow! Enter the Fist.
In the episode "Avatar Day", Sokka channels Judge Bao, with some Sherlock elements, while attempting to solve the case.
The signal horn that one of the Fire Nation sentries uses when he spots Aang in the episode 'The Blue Spirit' makes a sound precisely identical to the horn sound effect which kicks off the intro of the reality TV series Survivor.
Airbender arrows look a lot like the markings on the head of the Red Tornado.