The Non-Euclidian Staircase

I'll start with the favorite, a game I'm rarely without just in case I have 15 minutes and someone interested in playing. En Garde is a Knizia classic, a card game of fencing that, despite its simple and highly mathematical mechanics, achieves a very good feel in play. It's easy to learn, as the rules teach the game in three levels, starting extremely simple and slowly adding the more complex parry and advance-and-attack rules. It's a small game (just a couple dozen cards) with only a few "bits" (cards you lay out to create a board and dice printed with fencer silouettes to use as place-markers), but it packs a lot of fun into a small package.

Settlers of Catan is also an easy choice. Players are in the role of tribal leaders on an island trying to build up from a couple villages to a thriving urbanized culture more quickly than do the other tribes on the island, building with a set of 5 resources developed at random each turn. Trading between players is all but required, as another player will likely be the only easy way to get the resource you need. Some people dislike its high "luck" element (resources are produced at random and if your numbers don't come up, you lose out), some people make it take forever by playing too seriously (the style of game really calls for quick, light play), but if you can avoid taking it too seriously, it's a great way to pass some time. Almost anyone, even non-gamers, can quickly come to understand its goals and methods and become a reasonably good by their third or fourth game. And, if you keep the game quick, it plays fast enough that you can get through several games in an evening. The expansions are more for the pure gamer crowd (particularly Cities and Knights set), as they add complexities and length for strategic options that casual players aren't likely to appreciate, but the basic game is pure gold.