Zuko was one of the last characters to be written into the show. The Fire Lord was meant to be the shows main antagonist, but the writers realized he would not be able to cause Aang and his friends trouble sitting from his throne. Zuko was then written into the show.
The opening shot of every episode (the shot which contains the episode's number and title) is accompanied by a melody. The melody for any given episode is unique and never repeated exactly in any other opening shot. Often, this melody characterizes the episode's most significant event. For example, the melody in "The Painted Lady" is part of the Painted Lady's theme, and in "The Firebending Masters" - a slow version of the Sun Warriors chanting. The melodies of the first and the last episodes of the series are the different versions of the Avatar theme.
During the opening credits, five silhouetted figures are shown demonstrating various bending styles. Four are identifiable as characters from the show: the Waterbender is Master Pakku of the northern water tribe; the Earthbender is Sud, Roku's earthbending teacher, he was also an early prototype character design for Toph but that plan was scrapped; the Firebender is Princess Azula; and the Avatar who demonstrates all four bending arts is Roku - the Avatar before Aang. Although all four were present in the opening credits since the first episode, they actually make their first appearances in episodes 18, 46, 12, and 7 respectively (although a statue of Roku is shown in episode 3 as well). The Airbender is the only model never identified with a character in the series or supplementary materials.
Mako, who played Uncle Iroh, passed away in the middle of recording his parts for the last 7 episodes of season 2. They had to quickly re-cast the part of Iroh. The re-casting put a halt to production on season 3. Mako's part was succeeded by Greg Baldwin in the 3rd season.
Only five of the characters throughout the series have their age explicitly mentioned: Aang is 112 (though only 12 biologically), Zuko and Yue are 16, Toph is 12, and Tom-Tom (Mai's brother) is two. It is also mentioned that Mai is 15 years older than her brother, which makes her 17. Besides Aang, at least two other characters are over 100 years old: Bumi (knew Aang before he got frozen) and Guru Pathik (knew Monk Gyatso). Certain pieces of information from various episodes also imply that Firelord Sozin, Avatar Kyoshi, and Avatar Kuruk had lived for over 100 years each.
Katara's name was originally Kya in the unaired pilot episode, but was changed before the start of the series proper. However, Kya was used for the name of Katara's and Sokka's mother later in the series.
The calligraphy on the top and bottom of the map form a rhyming couplet when read aloud in Chinese. The characters at the top read "Powers are divided into four," while those at the bottom read "The world is guided by one."
The character of Admiral Zhao was inspired by the character William Tavington from The Patriot (2000). Casting director Maryanne Dacey was asked to find someone like the character and managed to get Jason Isaacs, who portrayed Tavington in the film.
Counting the episodes where they don't have any lines, the most prominent characters of the show are Aang, Katara, and Sokka, who appear in every episode of the series except episode 27 "Zuko Alone" (although Aang's spiritless body can be seen during the "Previously on Avatar..." introduction). The second place belongs to Momo, who appears in 56 episodes, followed by Appa with 54, Zuko with 48, uncle Iroh with 39, and Toph with 36 episodes. The most frequent recurring character is Azula, who appears in 22 episodes.
The name of the princess from the Northern Water Tribe, (and Sokka's first girlfriend) Yue, can be (depending on the tones) the Chinese word for 'moon'. And Suki, the name of Sokka's second girlfriend, (more commonly anglicized as tsuki) also means moon, only in Japanese.
The firebenders' duel, which plays an important role throughout the series, is called "Agni-kai". "Agni" is a Sanskrit word for "fire", and is also a name of a Hindu fire deity. "Kai" can be translated from Japanese as "meeting" or "together". "Fiery meeting" is an adequate description of the Agni-kai.
'Lake Laogai', the headquarters of the Dai Li agents, has a meaning in Chinese: Laogai is short for Laodonggaizao which means 'Reform through labor' and is the name of the Chinese political prison camps where many political opponents where incarcerated during the 60's, and is still in existence.
The trivia item below may give away important plot points.
Aang's Airbending teacher, Monk Gyatso, is named after the current Dalai Lama, whose full name is Tenzin Gyatso. It's also been confirmed that in the upcoming spin-off series, Legend of Korra, Aang has a son named Tenzin.