A lot of people ask why anime in particular? There are a lot of western cartoons which are excellent. A straight answer would be: I don't really know, so I'm going to try and distill it into a few points.
1. Anime, as opposed to any western cartoon, has a plot, as opposed to just a premise. A cartoon premise is something like 'deadbeat delivery boy has wacky hijinks in the future' (Futurama, incidentally one of the few cartoons I actually like as much as most anime series). Anime on the other hand has a story to tell, and a number of episodes to tell it in. This means that it can tell you the story, and doesn't end up dragging on well past its sell-by-date *cough*Simpsons*cough*.
2. The animation in anime tends to be of higher quality. Just compare a typical cartoon with a typical anime series, looking particularly at things like light shadow, perspective, how the human form is drawn, how colour is applied. In all of these the Japanese are light-years ahead of us in terms of sophistication.
3. Anime represents a whole new culture to unlock. 4-Kids may not like having anything vaguely Japanese in the anime they translate, but I love it. I love coming across something I don't understand in anime, be it a particular phrase, maybe a cultural signifier, an example of japanese cuisine, manners, holidays, festivals, technology. I love then diving into the internet and finding out precisely what that might mean to a Japanese person.
4. Anime is far less afraid that most cartoons. Partly this is a viewer thing. Most viewers don't want real violence, real nudity or real issues in their cartoons unless they are handled in a non-serious manner, like in South Park. Anime in that sense is far more daring.
5. Anime tends to have excellent music. I know this is a very personal thing, but listen to the soundtrack of a cartoon. Bland? Uninspired? Boring? Pretty much. Anime, on the other hand, can have incredibly good music.
Well, those are just a few reasons. There are cartoons that I like as much as some of my favourite anime series. Well, a cartoon, the aforementioned Futurama. However, in almost every way anime tends to be better than western animation.