Part of it was the fact that each episodes only about twenty-two minutes. In the time it would take us to watch a single episode of The Tudors or Buffy The Vampire Slayer, we could get through two or three episodes of Avatar. The plot was also very fast-paced, and while individual episodes were usually self-contained plot threads would return again and again throughout the series. Lots of variety in the story, too...some episodes were completely silly while others were quite serious, and even though it would have been really easy to fall into a repetitious "Aang meets bad guy, Aang fights bad guy, Aang learns something new" pattern the creators managed to avoid doing so.
I also *really* liked the attention to character design and detail. Usually in an animated series, characters will have only a few wardrobe changes. For example, take the anime Sailor Moon. Over the forty episodes of the first season, main character Usagi only has about five outfits that she wears: her school uniform, her 'sailor suit', pajamas, summer casual clothes, and winter casual clothes. Her hairstyle never changes. Her image doesn't change at all...and neither do any of her friends. This is pretty standard. The X-Men cartoon I watched in elementary school was much the same. Each character had his X-Uniform and one or two casual outfits. Nothing else. Now compare this to Avatar, where over the course of three seasons one character, Zuko, wears over twenty-five different outfits. (An excellent image that someone put together - probably to make one of those electronic paper dolls - showcases the variety really well.) His hair lengthens, gets cut, and changes hairstyle throughout the series. Most of the other main characters also show physical evolution - longer hair, growth in height, that sort of thing - as the series progresses. I guess it's a little thing, but I think it's really cool that this sort of effort was put into the visuals. I certainly wouldn't expect that in a show found on Nickelodeon.