This is...harder to answer than it should be. My taste is so eclectic sometimes I wonder what I actually do like. xD
But I think, generally speaking, I like anime with one of two things: a good narrative, or a strong cast of characters with strong chemistry between them. I need one or the other to keep me invested...both is ideal, but it seems anime these days rarely have both. With the cast though, usually I need a good villain, because villains drive the plot forward and protagonists are boring. A villain that either a) actually does meaningful things, b) wins, or c) does a believable heel-face turn is the best kind. I want someone unredeemable who I can cheer on whilst hating at the same time, or someone who goes through a solid redemption arc to become the best character in the show. If there isn't going to be a decent backstory to draw me in with a lack of characterisation - i.e. early stages of Gundam anime, especially ones like Iron-Blooded Orphans and Seed - then the characters need to be appealing right off the bat.
...that said, I will also settle for a large ham. The bigger the better. You don't need a complicated personality with believable motivation or even a character arc when you're JEREMIAH STORM OF MY MOTHERFUCKING LOYALTY GOTTWALD or your personality has been PASSED DOWN THROUGH THE ARMSTRONG FAMILY LINE FOR GENERATIONS.
Animation isn't a big deal to me...as long as it isn't disgusting like Eureka Seven's. I'll put up with a lot on this front, and I actually prefer older art styles for some things - I still think Char Aznab---sorry, I mean, Mobile Suit Zeta Gundam is the best looking Gundam to date. What IS important to me, though, is sound design. Any anime worth watching for me will have a strong soundtrack that will carry it through the more dramatic moments and give real weight to the mood the scene is trying to set. It's the difference between, say,
Smile Precure's transformation theme and
Star Twinkle's. The former is energetic and complements the animation perfectly, the latter is just a fluffy mediocre song that doesn't really add anything to the scene other than background music. A good song will really lend the right amount of weight to any given situation. Like every time Vogel Im Kafig plays in Attack on Titan you KNOW shit is about to go down and it's absolutely chilling, and Cross Ange is just filled with moments like this because of Akiko Shikata's amazing vocal work. But anime OST should not just be background noise, and any anime that treats soundtrack like just background noise is immediately less because of it. Most of my favourite shows have a very strong soundtrack.