The word "anime" is an abbreviation of the Japanese loanword "animeeshon" - from the original English, "animation." In Japan, it literally just refers to any animated material.
In North America, the term "anime" has been re-borrowed to refer specifically to animation that came from Japan. Therefore, the singular and defining quality that makes a particular animated show "anime" is that it was made in Japan. Granted Japanese anime have a very distinctive style, but that's not actually part of the etymology of the word.
The fact that some North American productions (like Avatar: the Last Airbender) deliberately copied the style of Japanese anime does not properly make it anime, at least in my personal opinion. That's just American animation studios trying to cash in on the "anime craze" (and perhaps succeeding, I don't really follow the statistics.)
If you choose to acknowledge things like Avatar as anime as well (which is entirely your right, btw), then you are creating your own definition for "anime" which no one else can really answer.
Edit to Note: There's no weaboo angle here - I'm not saying things which aren't anime are bad in any way. Avatar is a fine show, it's just not anime, as far as I'm concerned.