Inversely, in a game with incomplete information, players do not possess full information about their opponents. Some players possess private information, a fact that the others should take into account when forming expectations about how those players will behave. A typical example is an auction: each player knows his own utility function (= valuation for the item), but does not know the utility function of the other players. See for more examples.
Games of incomplete information arise most frequently in social science rather than as games in the narrow sense. For instance, John Harsanyi was motivated by consideration of arms control negotiations, where the players may be uncertain both of the capabilities of their opponents and of their desires and beliefs.
It is often assumed that the players have some statistical information about the other players. E.g., in an auction, each player knows that the valuations of the other players are drawn from some probability distribution. In this case, the game is called a Bayesian game.
Complete information is importantly different from perfect information. In a game of complete information, the structure of the game and the payoff functions of the players are commonly known but players may not see all of the moves made by other players (for instance, the initial placement of ships in Battleship); there may also be a chance element (as in most card games). Conversely, in games of perfect information, every player observes other players' moves, but may lack some information on others' payoffs, or on the structure of the game. A game with complete information may or may not have perfect information, and vice versa.
Games of incomplete information can be converted into games of complete but imperfect information under the "common prior assumption." This assumption is commonly made for pragmatic reasons, but its justification remains controversial among economists.