I've played a lot of them. In a family with five kids and twice that many close cousins, playing lots of board and card games is to be expected.
So I thought I'd run through a handful and briefly describe what features made each of them fun. The idea was brought on by Brian's ongoing community project.
Monopoly is, first and foremost, a great mix of luck and skill... with heavy emphasis on luck. Not only are your dice rolls and card draws random, but the order and timing of those rolls and draws matter as well. Monopoly also allows for personal strategy. Some people like the utilities or railroads, some the expensive properties, others the cheaper properties while are easier to build on. The game encourages verbal interaction through trading and rent payments. Though I'd guess that most Monopoly players have only played the original version, it's significant that the gameboard and cards can be tailored to different cities, universities, and other sources of pride.
Since Sorry!, many boardgames have used its basic premise of one player sending another back to an earlier position. Making one player the cause of another's setback is a great way to encourage social interaction. It's more fun when some reciprocity is all but assured. Later games, like TriBond, spuns the concept of setbacks so that one player could forcibly switch places with another. This was a great idea, because it means no player ever falls back so far that he or she no longer has a chance at winning. It's generally not fun staying in a game when you know you can't win.
Yes, it's a word game, but the basic gameplay is applicable beyond language. Scrabble is about building something out of randomly distributed parts. There are strategic elements, but it's mostly about seeing potential in a mess (Boggle plays off the same concept). Ironically, finding potential order in a mess often feels like creativity, even if all you've done is replicated words everyone already knows. Minor creativity is fun for all, but more heavily creative gameplay (like in Balderdash) has a smaller audience.
The war simulation game Risk emphasizes strategy. Unlike Chess or other strategy games of old, it involves enough chance to ensure variation between games. The beauty of Risk is how dependent gameplay is on individual players. Four or five different personalities with adaptive strategies mix to create a unique playthrough every time. Like Carcassonne, the player must make a strategic decision every turn without the benefit of knowing what the gameboard will look like the next time it's his or her turn.
The king of poker games is like Monopoly in that it's a strong blend of strategy and luck, but it's defining feature is bluffing. More than any other game, Five-Card Draw encourages players to hide their luck from other players (and, of course, reveal it once the round is over). Betting allows players to select their own levels of risk, and one player can pressure another into greater risk. Other games involve not knowing opponents' strategies, but Draw involves not knowing the opponents' resources as well.
100% luck, yet still fun. Each player gets seven cards blind (face-down). Each turn, you flip over your cards one at a time until you have a poker hand that beats the player before you. Turns continue until all cards are turned over. What makes this game great is the open betting. There can be as many as a dozen or more turns of betting, and each time you can bet however much you want... the others must match the bet or fold. The pots can be enormous, you're betting purely on faith in luck, and the game is full of intense anxiety as the cards are overturned.