The term comes from Japanese, and in that language, yes: Avatar is anime. As used in Japanese, the term simply refers to animation, and does not distinguish it by nation of origin. But as a loanword in other languages, however, Avatar usually doesn't qualify. Most languages other than Japanese have borrowed the term anime to refer specifically to animation of Japanese origin.
Anime-as-loanword is a somewhat fuzzy term, but generally refers specifically to the animation's origin, and not necessarily where it is produced. Much of what we would call anime is actually produced outside of Japan: it originates there, but the production is outsourced to studios in other countries. Kaleido Star is an example of this: its origin is Japanese, but much of the work was done as a collaboration between Japanese studio GONZO and Korean studio G&G Animation.
Likewise, a surprising amount of animation that we wouldn't call anime originates outside Japan, but the animation work is outsourced to a Japanese company. ThunderCats is an example in both its incarnations: few people outside of Japan would call it anime, but the animation work was done there (the 1980s TV series was done by Pacific Animation, while the recent series was done by Studio 4C).
Avatar: The Last Airbender represents a third category: it is neither of Japanese origin nor production, but is so heavily influenced by anime tropes that it can be difficult to tell the difference without researching the show's credits. The creators are American. The animation work on the original series was outsourced to a number of Korean companies. A few episodes of its sequel series, The Legend of Korra actually were animated in Japan (by Studio Pierrot), but most of the animation work on this series was done by a Korean company: Studio Mir. But despite the franchise's origin and production, the influence of anime on it is extremely strong, as its own creators admit. Anime's influence can be seen in virtually every aspect of the franchise, from its visual styling to its world-building to its character depictions.
Whether this makes it count as anime, or at least as "close enough", is a subject of debate among some fans. In the strictest sense, it doesn't qualify unless you're speaking Japanese. But if it's indistinguishable from anime without closely examining the credits, then perhaps there isn't much sense in insisting on the distinction. So, at least, say the people who would include it. There are also fans who agree that it's "not technically anime," but consider it close enough that it should be included in anime fandom anyway ("honorary anime" is a term you hear from this camp sometimes).
Even in Japan there is some debate, in a case of what can only be described as linguistic ping-pong. It's common knowledge that the term anime was borrowed (in abbreviated form) from English, to mean animation. English then re-imported the term, referring specifically to anime of Japanese origin. And although it has yet to really hit the mainstream, some Japanese fans have taken to re-re-importing the term, this time in the unabbreviated form animation, to refer to animation of non-Japanese origin.