In addition to playing a standard game requiring a score of 301, 501 or whatever there are a number of other variations of games that can be played on a darts board.
One possibility which is very good for practicing your aim is round the clock. In this game the players (any number of people can play from one up) take it in turns to throw three darts. The aim is to hit each number of the board in numerical order, starting with one and working round to twenty, and then the bull. Once one number has been hit the players moves on to the next. If a player hits three sequential numbers in their turn then they get another go. More difficult variations on this theme include requiring three of each number (counting doubles and trebles as two and three respectively) or requiring doubles instead of singles.
A good game for many players (between 3 and 20) is killer. As the name suggests the aim is to become "killer" and eliminate all the other players. After deciding playing order (usually by each throwing one dart and playing in order of how close the darts go to the bull) each player throws a single dart at the board left handed (left handed players throw right handed). Whichever number the player hits becomes their number. If a player misses or hits another player's number then they throw again and continue to do so until they hit a number that hasn't been taken.
The players then take it in turns to throw three darts. Each time they hit their own number they gain one life. Once a player acquires three lives they become killer and may aim at other players. Each time a killer hits another player they lose a life. Doubles and trebles count as two and three lives respectively while the bull (if it isn't a players number) counts as one life for a non-killer or one life of everyone else for a killer. If a player's number is hit by a killer when they have no lives they are dead and removed from the game. Generally players who haven't yet had a throw are immune from being killed.
The most amusing aspect of the game is the selection of targets. Very few players play to actually win instead playing to survive or aiming at people to settle vendetas left over from earlier games. Due to the nature of the game it is easy for a good player to be easily killed by two or three much weaker players and so good darts players rarely win games of killer, in fact it's quite common for poor players who aren't seen as a threat to win. If all players are playing competitively killer is an interesting experiment in game theory.
A more difficult and complicated variation is that if a killer hits themselves then they lose a life while if a non-killer hits another player they gain a life. This can be particularly irritating when another player has the number next to you and you accidentally lose yourself three lives while aiming at them.