I am pretty much all over the map as far as game preferences go. That's probably because I am still learning what games I like and dislike. I have learned a few things, though.
The thing I appreciate most in a game is elegance. I love it when simple rules lead to deep, complex, or maybe just fun situations. I consider Through the Dessert and Battle Line to be prime examples of this. Pitch Car and/or Carabande fit as well, but their simplicity is fun, not deep.
When I first started out, I was pretty convinced that I didn't like abstract games, but I have learned that I love several nearly abstract games. Theme is important to me, but not essential. Tikal is probably the game that attracted me most with theme, and then sucked me in with interesting game play. I also apparently love miniature games, but object to collectibles and don't like to paint. When I do get a miniatures game, I don't get to play them. Doom has shown some promise, though.
I don't feel that a game has to be highly rated for it to be the perfect game for the moment. Theme is usually the driving force for games like this. For example, Formula De is a great game that I am not always going to want to play. It is however the perfect game when the mood is right. Other games that sometimes fit the mood are When Darkness Comes and Lord of the Rings. Of course LOTR is a great game any time, but it usually comes out for a reason.
I don't mind randomness and chaos, but in longer games, they need to manageable. In a short game, nearly anything goes, but luck and chaos are best when they lead to interesting surprises. I am one of the seemingly few people who like Fluxx (or at least it seems that way sometimes). If I overplayed the game, or experienced some of the unending games that others have reported, I might like it less, but those negatives just aren't hapenning to us. I also really enjoy a quick game of Cloud 9. This risk taking game of "Chicken" is great fun. Games like Maharaja and Dos Rios are not out of control, but can lead to some wonderful "Wow! I didn't expect that!" moments.
Finally, I often to like resource and building games. Building toys were always favorites as a kid, and I guess that need is still inside. Settlers of Catan, San Juan, Attika, and Carcassonne all fit as both building games, and as games with manageable luck. Unfortunately, in Settlers, the luck is almost never a happy surprise, so I guess I like the building, because I definately like Settlers.