Mate With King and Queen Vs King

This teaches you how to quickly and efficiently checkmate your chess opponent with only a King and Queen vs. his lone King.

Steps

Move your Queen to a point that is a Knight's move away from the enemy King (A knight moves in an L-shape).
Every time your opponent moves, move your queen in that same direction 1 square.
If your queen is on the edge of the board and cannot move in the same direction as the opposition king, move two squares across so it's still an L-shape away from the king.
When the enemy king is on the edge of the board, cut off the king so that it only has one row on the edge of the board on which to move.
Move your king up, and move the queen so it checkmates the enemy king while also being protected by your own king.

It is also possible to use this method with a rook and king, using the following steps.

Confront his king with your king, keeping each other one space away, and move your queen/rook to that line.
Move your queen/rook down one space, and he will be left with no other choice but to move backwards.
Continue this process until the back row.
After you reach the back row, use it one last time, and presto! checkmate!

This method is more complicated than the other methods, but is faster.

Use your queen to create a surrounding wall. Use the wall closest to your king.
Time queen moves and king moves to gradually force the king to the edge.
Once the king is at the edge, move your queen to stop the king escaping from the edge.
Maneuver your king two squares ahead of the opponent's king.
Move your queen one square above the opposing king, where it is protected by your king. Checkmate!

When you trap the enemy King, make sure you don't stalemate him (leave your opponent with no King moves)! Not only must the enemy king not have a single possible move, but he must also be attacked at the same time, or else it's a draw! If you are playing an official game, the 50 move rule applies, which states that the game is drawn in 50 moves by white and black without captures or pawn moves. Two kings alone is also a draw, this time by insufficient material to mate. Do not let the enemy king take your rook/queen, but be aggressive in forcing your opponent's king to the edge of the board.