Personally, I like solving the puzzles that are sort of logical - not so obvious that you get them straight away, but also not completely random (I wasn't keen on Treasure Island Dizzy because it felt like lots of the puzzles were random). I also like things like quicksand and so on where you have to keep jumping. And I like not having to worry too much about losing lives - like if there are plenty of extra lives, or you can eat food to give you a boost - for me the fun is in solving the puzzles, and I get a bit annoyed if it's too easy to die!
I really HATE looking for the coins/stars/whatever though. I also dislike games where the map is really spread out, and you have to haul loads of items about from one side to the other, and ones with awkward creatures to avoid - like a spider that's a wee bit too close to the edge so you keep walking into it.
What does everyone else think?