Well, I'm a relative newbie when it comes to anime, though I am slowly making my way towards otaku-hood. My first anime (outside of Pokemon, Yugioh et al.) was Naruto, about 8 years ago, and since then I've been hunting down as many anime as I could.
What attracts me to anime? It's hard to say, though it probably boils down to the creativity, aesthetics and sentimentality that you'd be hard pressed to find in Western animations outside of older Disney movies and Pixar (which also happens to be CG). Honestly, when was the last time you saw something this pretty in a hand-drawn Western cartoon?
As for my taste in anime, mostly anything goes. Shonen, mecha, dramas, even moe-blob slice-of-life series--as long as the plot and characters are nice and there's plenty of content to keep me wanting to watch. Of course great visuals and music are always a plus. Currently I'm trying to catch up on fairly recent good animes that I've heard about (TBH, most of the anime I've seen were made after 2000), though I'll try to get into classics sometime soon. At the moment it's somewhat difficult to break from the sleek, polished animation of modern animes :D