Different games have different rules. The two most popular in North America, the traditional x01 games and American Cricket, are discussed in this FAQ, the rules to other games and variants are provided in part 2 of the FAQ.
For the uninitiated, "x01" refers to the number from which a game is started: 501 and 301 being the most common.
In general, each player has a turn or throw consisting of three darts. To determine who will play first, each player (or one player from each team) will throw one dart at the bullseye. The player who's dart is the closest to the center of the board plays first. Often the order of play is determined by a toss of a coin instead. This procedure of throwing at the bull to see who plays first is often called "diddling for the middle", "one up" or "bull up". In the event of a tie throw, both players must throw or "diddle" again in reverse order. If both darts land in the same area of the bullseye (the outer or inner bull), it is considered a tie even though one dart may be noticeably closer to the center of the dart board.
In competition, it is common to allow each player a warm up of three throws (9 darts), taken one throw at a time in alternating turns, before a match begins.
In a game, darts not landing in the scoring area do not count score but do count as darts thrown and may not be thrown over. However, in the event that a player drops a dart not having made an effort to throw it, he may pick it up and throw it. A dart's score is determined by the point of entry of the dart based on the wire divisions, not necessarily the colour.