What's "fun" anyways?
That game was fun. We had fun playing it. We had so much fun, we didn't even realize it was already 3am. I had fun the next day trying to work on 4 hours sleep. Want to play again tonight?
What aspects of gaming give you pleasure? For me, each game I play satisfies different areas of my brain. The games I want to play depend on my mood, who I'm going to be playing with, how much time we have, the time of day, the phase of the moon, and the current terror alert color.
Social interaction with family and friends?
I enjoy getting together with both, and gaming is a good excuse. However, in other than severely light games or party games, a lot of the normal social interactions do not exist, as players are busy thinking about their next move. In these games, you can talk about anything while playing.
Games: Apples to Apples, Coloretto, TransAmerica, Uno
There are also games, by their very design, that force in-game social interaction. However, you won't spend much time talking about Aunt Martha's new cat.
Games: Diplomacy, Traders of Genoa, Werewolf
Wits and chicken?
Sometimes players need to not only evaluate what the game situation is, but also what others are likely to do. You can make choices based on how well you know the other players, based purely on the facts available, or simply guess. In some of these games, you are playing chicken--seeing who will "jump" first.
Games: Citadels, Medina, Meuterer, Ra
Figuring out what to do with ever-changing and potentially random situations?
I like playing by the seat of my pants. Intuition is sometimes more enjoyable than analysis for me. Games that defy analysis often have shorter playing times. This can be accomplished with bluffing mechanics, minimizing choices, or having enough randomness.
Games: Carcassonne, Ra, Ticket to Ride, Ys
Finding the right combinations?
In games with many possible actions or many targets upon which to perform actions, finding reasonable combinations is a great challenge.
Games: Domaine, Euphrates & Tigris, Puerto Rico, San Marco
Crunching the numbers? Reading 20 moves ahead?
Sometimes I just want to make my brain burn with like a large serving of cerebral habaneros. I may not look like I'm having fun--sitting there with my head in my hands--but trying to out-think an opponent in a deep and heavy game is very fulfilling.
Games: Go, Bridge, Chess
Some games aren't as cruel as the above but ultimately fall into the same category.
Games: Samurai, Santiago, Torres, Through the Desert
It's fun to beat up on your opponents, and take revenge for getting beat up. It doesn't have to be a wargame to have the same "Take that!" feel.
Games: Citadels, Kremlin, Nuclear War