There's a lot I enjoy. It depends on the show though as anime has taken on such a broad term anymore.

Generally though I enjoy the animations. And I love the character designs, be them insane or not, it's just nice to see something interesting that characters wear that you'd never see in real life except outside of fashion shows or anime conventions.

I also enjoy how watching the Japanese versions of the shows the dialog can get pretty poetic at times. And in my opinion the characters and stories are much more developed then so many American forms of media. Anime also doesn't try to be gimmicky or follow as many social norms as American media does. Anymore American shows and movies will just sell out and do something 'cool' because they think it'll get them better ratings or what not. Really most animes are better developed then so many other forms of media now a days.

I also like how sometimes different series' will actually surprise you and aren't as formulated as other shows and movies I watch now a days. I also enjoy how there's a lot of animes I watch to the end and they leave me pondering over it for days, weeks or even years (damn you Evangelion!).