Tile Laying Games
Tile laying games require players to draw single or multiple tiles and play them onto the board. Build and develop the land and peoples one tile at a time. Tigris and Euphrates is a game of this type designed for at least three players.
Cooperative Games
Unlike traditional games that pit players against one another, cooperative board games have players join forces against a common enemy. There are a variety of quasi role-playing games that fall into this category wherein the players seize a dungeon or castle to overthrow some wretched evil that wants to destroy man kind. "Shadows Over Camelot" is a three- to seven-player game that fits this category.
Resource Management Games
Resource management games require players to manage the resources they collect to build an empire and win the game. Resource distribution is usually limited and trading is often required. Use diplomacy, strategy and hope for some luck to win at resource management games, of which "Settler of Catan," a three- to four-player game, is a prime example.
Auction Games
Auction games require players to utilize bids to acquire game pieces that work toward winning the game. Auction games often have buying, sellin, and trading components while also requiring players to strategically use their limited bids. You can try "Modern Art," a three- to five-player auction game, for a taste of the genre.