This terminology simply refers to the number of balls in play in a group that is being wagered upon. Each player has her/his own ball.
Two ball is another way to say a head-to-head matchup. Three ball denotes there are three players in the group.
To win either type of bet, you must correctly wager upon the golfer with the lowest score of the group. In these types of bets, where the player finishes among the final standings doesn't really matter.