What characteristics tie anime styles together?

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.

I wouldn't say there is a particular set of characteristics that tie them together. But if I had to point out ONE particular characteristic that is widely used in the anime industry, are the eyes.

In many cultures, the eyes were considered the path to the soul, and manga uses that in many levels in their characters. Usually in manga/anime, you can tell entirely what is going on with a character by their eyes, and many times, you may tell exactly what is that character's personality.

In anime, the eyes of a character may be bigger or smaller, but they will almost always be filled with personality.

Ok. It's funny when you bring Avatar:tLA. Because it is western production made in Asia. There are plenty of animations that are financed by western companies but made in Korea (French like to go to the Best Korea for the prices). Korean animators are strongly influenced by Japanese style. Which came mostly from Osama Tezuka Astro-boy. It's aesthetic are the first sings of what would be later called "Anime".
Apart from the easily notable style what distinguish western animation from Japanese/Asian one is pacing. Mangakas adapted different storytelling because they had a lot of paper on their hand. Compare 120 page of monthly manga to 48 pages (with ads) comicbook.
Later, they carried that style to animations. That's why you had 12 episodes of Tsubasa running with a ball or 24 episodes of Goku charging and attack.

So, back to Avatar. It have Asian/Japanese aesthetic in character desing but due to western producers it's time pacing is very condensed and fit into 20 episodes compared to circa 40 episodes a year of Dragon Ball Z.

For a long time anime was considered just animation made in Japan. Today there is a lot of discussions actually about it with american cartoon taking a lot from them and terms like "chinese cartoon" no longer being a joke. But anime doesn't have, and shouldn't have, specific characteristics you are asking about. Otherwise famous works like Ping Pong The Animation, Mind Game, Mononoke and many others would no longer be considered anime.