If a player still has a pawn, this pawn can be promoted to a greater piece (Queen, Bishop, Rook, or Knight -- usually the Queen is chosen) and, if the queen or rook is chosen, this means a win for the player starting with the pawn. Stalemate only occurs if the pawn is captured by the opponent before it can be promoted. Also, if a player is left with only a king and knight, or only a king and bishop, and the opponent only has a king, or a king with either (a bishop or knight), in these situations, it is impossible to bring the opponent to checkmate. Only with the king and a rook or (obviously) with the king and a queen, can checkmate be accomplished.
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stalemate occurs when there are very few pieces left on the board and checkmate can not occur. The kings must be at least one square apart, so if that is the only move your opponent has achieved checkmate and has won