Basic Rook Checkmate
This is an important checkmate to know and it's rather easy to learn. The themes apply in most K-R endgames.
In short, a K-R checkmate can be summed up with the following procedure, outlined in the above example:
1. Trap the enemy King in a box, cutting off as many squares as possible with the Rook.
2. Bring up the King to support the Rook.
3. Force the Kings to be a "Knight's Move" apart, leaving two possibilities: a) the King approaches and the Rook checks.
b) The King retreats and the position is repeated.
4. Once the enemy King is on the back, trap it there with the Rook.
5. Repeat Step 3a to force the checkmate.
This may not be the quickest method, but it works every time.