It varies for me. In romance anime, I want characters that I can actually get behind and like, when things go bad for them, I want to feel bad, when they're happy, I wanna be happy. I want good obstacles in their path and have them eventually end up happy, or if it can be done really good, have it be tragic. Not too many anime do that for me, though RomeoXJuliet is the first in years that has been able to do that for me, easily my favorite anime of the year so far.
For comedy, I like slapstick when done right, like in Keroro Gunso, one of the funniest anime in years, and Hayate no Gotoku to a lesser degree, they're both really funny anime for different reasons. Keroro Gunso's still my all time favorite comedy anime though.
Outside of that, I just like seeing an anime be creative or give me interesting characters, and if it's not that, then just something that can reinvigorate a tired genre and character type.