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..
The same episode also has an obvious one in Aang's first two nightmares.
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 Tattooine. 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.
The name, and the character, is a reference to The Rock.
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.
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 in the Hayao Miyazaki movie "Spirited Away".
The shout out to Miyazaki's works happens again 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.
"Avatar Day" has a dead person speaking through someone whose alive at a trial to give their side of what happened at the crime. Rashomon anyone?
Ba Sing Se's political landscape is very resembling of 1984's.
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.
In Sozin's Comet (Part 2, I believe?), one of the people June soundly trounces in the bar may or may not be Ryu from Street Fighter.
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.