I enjoy games that involve words or categorizing. For example, my friends and I like to play a drinking game called "Sex, Drugs, and Rock n' Roll", where we all go through the alphabet naming something related to either sex, drugs, or music...so we all start on A until someone messes up, then move on to B, etc. I also like Taboo and that game "Catch Phrase", where you have to get the rest of your group to guess a word/phrase without you actually saying it.
In terms of video games, I like fantasy/RPGs. Final Fantasy is always good, and I saw someone mention Shadow of the Colossus, which is one of my favourite games besides Diablo II. I just finished a Japanese game called Persona 4, which I also loved. I really enjoy games where I can assume the role of another character and relate to him or her on any level.