Opening
Sometimes it is possible to checkmate within the first few moves of the game. This can generally only be achieved if you are playing a very inexperienced opponent; to be checkmated in the first 10 moves, you need to be very careless and leave your king open to attack. For example, if white moves the pawn to the right of the king up one space, and black moves pawns up to allow the queen and bishop to move diagonally to the right, white's king can be checkmated after just four moves.
Mid-Game
Among capable chess players, checkmate may be achieved in mid-game. The possibilities here are endless. Supposing the two players have most of their pieces still intact 30 moves into the game, checkmate can be achieved with relative ease. One classic mid-game checkmate involves a boxed-in king. Suppose one player's king is located on the rightmost square in their back row (h1 for white, a8 for black). If the other player can move a queen or rook up so that it is threatening the king, then the king cannot escape, and is in checkmate. On the other hand, sometimes one player can establish a powerful advantage by mid-game by wiping her opponent's pieces off the board, and then pushing their king into a corner.
Late Game
Among experienced players, checkmate will usually occur late in the game with relatively few pieces on the board. In this situation, possibilities for checkmate are endless, since by this point every piece in the game has been moved several times and board positions may be very complex. In the end game, it may be possible for one player to get a pawn (a low-value piece) all the way across to the other side of the board and trade it in for a queen (a high-value piece). If both players have only pawns and kings, and one player manages to get his pawn across first, then that player may achieve checkmate by boxing the king into a corner with his queen. Other times, end-game checkmate is achieved by pinning the king into position with a rook or bishop, and then moving a knight in for the final check (checkmate).
Basic vs. Complex Checkmates
In addition to where they occur in the game, checkmates can be categorized by the complexity of their formation. Basic checkmates are achieved with just one or two pieces. If, for example, your opponent has a king in her back row and you have a rook in the same row and another rook one row down, then you have your king in a basic checkmate. If, however, you form an elaborate set-up where your opponent's king is trapped between five or more of your own pieces, then that is a more complex checkmate, such as a smothered mate or Anastasia's checkmate.