I like most kinds of boardgames, esp. those with role playing, civilization, development and trading but I am not particular interested in war game, area majority, or trivia games.
I don't like dry or static games. It's boring that the same steps, actions or strategies appear in every play of such games. Games should provide fun, not just a mind or body exercise. Thus, I prefer games that facilitate interaction between players so that we can talk and laugh together. I also prefer games providing different kinds of thinking or strategies in different stages. I like games to be educational as well - they come with a healthy theme or players can practice their logical thinking and social capability. Finally, I like games with a matching theme.
Random games are not my cup of tea. However, I am not all after controls. I think games should have some random or hidden factors to have fun. Sometimes, perfect information games may be boring. I don't mind chaotic and luck-dependent games if they provide loads of fun.
I own and play many 2p games with my wife and my friends at leisure time. It's an easy way to pass time with your wife at home
If I had to keep only then board games in my collection, the will be:
Settlers of Catan
Descent: Journeys in the Dark
Lord of the Rings
Ticket to Ride: Europe
Tigris & Euphrates
Age of Steam