Chess/The Endgame/King and Rook vs. King
This mate takes longer to do than king and queen vs. king, because the king and rook have to work together to trap the opposing king on the edge (often in the corner). The most commonly taught technique involves confining the opposing king into a box using the rook, which is protected by its own king. Then, the box becomes smaller and smaller until the king is forced into the corner. From the diagram on the right:
1.Kc2 Ke5 2.Kd3 Kd5 3.Ra4 Ke5
After the king's position is improved, the box can be constructed:
The box is d8-d4-h4.
4...Kf5 5.Re4 (see diagram)
Now it's e8-e4-h4.
The box can't be reduced immediately, so the king creeps closer.
6...Kf5 7.Kd5 Kf6
7...Kg5 8.Ke6 Kg6 9.Rf4 Kg5 10.Ke5 Kg6 11.Rf5 Kg7 12.Rf6 Kg8 13.Kf5 Kg7 14.Kg5 Kh7 15.Rg6 Kh8 16.Kf6 Kh7 17.Kf7 Kh8 18.Rh6# is similar.
Back to shrinking the box: the process repeats itself. Notice that just as in K+Q vs. K, no checks are necessary until the actual mate.
8...Kf7 9.Re6 Kg7 10.Ke5 Kf7 11.Kf5 Kg7 12.Rf6 Kh7 13.Rg6 Kh8 14.Kf6 Kh7 15.Kf7 (see diagram)
Something important to know: if it somehow transpires that White reaches this position but it's his turn to move, all he has to do is move the rook anywhere along the sixth rank (except Rh6+ of course!). Then, Black's king is forced to move to h8, and White gives checkmate with Rh6#.
The alternative method is based on the opposition of the kings (see below) with the rook being far away. From the same starting position as before:
1.Kc2 Ke5 2.Kd3 Kd5
Once again, White must improve the position of his king first. Now, once the kings are in opposition, he checks along the fifth rank to push the black king back.
3.Ra5+ (see diagram)
Black has no choice but to give ground.
3...Kc6 4.Kd4 Kb6
4...Kd6 5.Ra6+ immediately cuts Black off the sixth rank.
5.Rg5 Kc6 6.Rh5 Kd6
6...Kb6 7.Kc4 Kc6 8.Rh6+ is the same sort of thing.
7.Rh6+ Ke7 8.Kd5 Kf7 9.Ke5 Kg7 10.Rb6 Kf7 11.Ra6 Ke7 12.Ra7+ Kd8 13.Ke6 Kc8 14.Kd6 Kb8 15.Rg7 Kc8 16.Rh7 Kb8 17.Kc6 Ka8 18.Kb6 Kb8 19.Rh8#.
Using either method, it should take about 15-20 moves to deliver the mate. Theoretically, it should take no longer than 16 moves to checkmate, but depending on the position, one might have to use a mixture of the two techniques to do this.