Depending on the anime, I usually like deep and powerfull characters.
My all time favorite character is Ghost in the Shell's Major Motoko Kusanagi - she is very good at what she does, with a history behind her. Other (favorite) "strong" characters would include the Count of Monte Cristo (he just looks so cool!) and Guts from Berserk.
I love seeing an insanely powerful character - one that keeps suprising the viewer. Guts was / is just insane. The Count's vindictiveness was amazing. And the Major's just so good at her job.
Other characters (a.o. in no particular order) would include Saber from Fate/Stay Night, Lucy from Elfen Lied and even Youko from the 12 Kingdoms (tragic heroine who discovered her hidden strength).
I don't particularly like the "pure yet simple" heroes (DBZ's Goku's idiotic innocence irritated me), but then Bleach's Ichigo would probably be my fav in a not-so-serious catagory. Then I just like any character who can kick butt in a humorous way. And of couse, being male, I tend to be biased towards female characters ^^