Mate with King and Rook against a King

The easiest way to learn is to remember the checkmate position:

[Event "checkmate"]
[Site "?"]
[Date "2012.??.??"]
[Round "?"]
[White "KR"]
[Black "K"]
[Result "1-0"]
[SetUp "1"]
[FEN "R5k1/8/6K1/8/8/8/8/8 b - - 0 37"]
[PlyCount "0"]

* 1-0

You get checkmate at the edge of the board. You can force the defending King to the edge of the board by using the same idea:

[Event "Forcing position"]
[Site "?"]
[Date "2012.??.??"]
[Round "?"]
[White "KR"]
[Black "K"]
[Result "1-0"]
[SetUp "1"]
[FEN "8/8/6k1/R7/6K1/8/8/8 w - - 0 23"]
[PlyCount "3"]

23. Ra6+ Kf7 24. Kf5 1-0

Black is forced backwards.

You can get to this forcing position by chasing the enemy King with your King:

[Event "Chasing the King"]
[Site "?"]
[Date "2012.??.??"]
[Round "?"]
[White "KR"]
[Black "K"]
[Result "1-0"]
[SetUp "1"]
[FEN "8/8/4k3/R7/2K5/8/8/8 w - - 0 19"]
[PlyCount "10"]

{Chasing the King:} 19. Kd4 Kf6 20. Ke4 Kg6 21. Kf4 Kh6 22. Kg4 22... Kg6 {
Kings opposite - ready for a check!} 23. Ra6+ Kf7 1-0

You don't get to the forcing position if it's you that puts the Kings opposite, because it's Black's move, and they will move out of the way, and you end up doing a silly dance;

[Event "Silly dance"]
[Site "?"]
[Date "2012.??.??"]
[Round "?"]
[White "KR"]
[Black "K"]
[Result "1-0"]
[SetUp "1"]
[FEN "8/8/8/4k3/R7/5K2/8/8 w - - 0 11"]
[PlyCount "8"]

{To avoid the silly dance...} 11. Ke3 Kd5 12. Kd3 Ke5 13. Ke3 Kd5 14. Kd3 Ke5 1-0

The way around this is to waste a move with your Rook. That way it's Black's move, and the chase is on again!

[Event "Losing a move with the Rook"]
[Site "?"]
[Date "2012.??.??"]
[Round "?"]
[White "KR"]
[Black "K"]
[Result "1-0"]
[SetUp "1"]
[FEN "8/8/3k4/7R/2K5/8/8/8 w - - 0 18"]
[PlyCount "11"]

{Don't do the silly dance here....} 18. Ra5 {'Lose' a move with the Rook!}
18... Ke6 {Chasing the King:} 19. Kd4 Kf6 20. Ke4 Kg6 21. Kf4 Kh6 22. Kg4 22...
Kg6 {Kings opposite - ready for a check!} 23. Ra6+ 1-0

So, putting it all together:

[Event "Putting it all together"]
[Site "?"]
[Date "2012.??.??"]
[Round "?"]
[White "KR"]
[Black "K"]
[Result "1-0"]
[SetUp "1"]
[FEN "8/8/8/4k3/8/8/8/K6R w - - 0 1"]
[PlyCount "73"]

{The worst starting point for White. You have 50 moves to mate.} 1. Kb2 Kd4 2.
Rh3 Ke4 3. Kc2 Kd4 4. Ra3 Ke4 5. Kd2 {Chasing the King} 5... Kf4 6. Ke2 Kg4 7.
Kf2 Kh4 8. Kg2 8... Kg4 {Kings opposite: ready for a check!} 9. Ra4+ Kf5 10.
Kf3 Ke5 11. Rh4 {'Lose' a move with the Rook!} ({To avoid the silly dance:} 11.
Ke3 11... Kd5 12. Kd3 Ke5 13. Ke3 13... Kd5 {'Lose' a move with the Rook!})
11... Kd5 12. Ke3 {Chasing the King} 12... Kc5 13. Kd3 Kb5 14. Kc3 Ka5 15. Kb3
15... Kb5 {Kings opposite: ready for a check!} 16. Rh5+ Kc6 17. Kc4 Kd6 18. Ra5
{'Lose' a move with the Rook!} 18... Ke6 {Chasing the King:} 19. Kd4 Kf6 20.
Ke4 Kg6 21. Kf4 Kh6 22. Kg4 22... Kg6 {Kings opposite - ready for a check!} 23.
Ra6+ Kf7 24. Kf5 Ke7 25. Rh6 {'Lose' a move with the Rook!} 25... Kd7 26. Ke5
Kc7 27. Kd5 Kb7 28. Kc5 Ka7 29. Kb5 29... Kb7 {
Kings opposite: ready for a check!} 30. Rh7+ Kc8 31. Kc6 Kd8 32. Ra7 {
'Lose' a move with the Rook!} 32... Ke8 33. Kd6 Kf8 34. Ke6 Kg8 35. Kf6 Kh8 36.
Kg6 36... Kg8 {Kings opposite: ready for a checkmate!} 37. Ra8# 1-0

Practise against a friend or a computer!

Chess Quotes

[cool blue cat says:]

COOL TIP: What does Reti mean? He was one of the strongest players of his day, but surely grandmasters are superb calculators! Yes, they are, but often they do not need to calculate something from scratch, because they recognise the type of position they are in, and they know what to play in that sort of position. All the calculating has been done before by someone else, and once you are shown how it works, you can use it and apply it in your own games.
— Dr. Dave.