Being up a rook in the endgame, even if your opponent has a pawn as compensation, usually means an easy win. However, some such endings might need a little bit of technique. One of the major ideas is pushing your opponents king toward an explosion. Consider:
8/8/6pp/6PP/8/8/3K3k/5R2 w
White to Move and Win
Here, White can't trade a pair of pawns because Black would then trade the other pair for a drawn king and rook vs. king endgame. Also, Black can't trade a pair of pawns since White would then push his remaining pawn toward queening. Play begins: 1. Rg1 Kh3 2. Rg2 Kh4. Now, 3. Rg3?? would be a funny stalemate, so: 3. gxh6 Kh3 (or g5) 4. h6 [1-0]. White queens and easily wins.
Next, consider:
7k/2R5/8/6p1/5p2/5P2/6P1/4K3 w
White to Move and Win
Here, White stalemate's the Black king (leaving the pawns free to move) in order to easily achieve a passed pawn: 1. Rg7 g4 2. fxg4 f3 3. g4 f2 4. Ra7 [1-0].
This "stalemating technique" can work wonders. Consider:
K7/5pR1/8/8/8/8/8/7k w
White to Move and Win
White wins like this: 1. Rg2 f6 2. Kb8 f5 3. Ka8 f4 4. Kb8 f3 5. Ka8 f2 6. Rh2+ Kg1 7. Rh1+ Kg2 8. Rh2+ [1-0].
Knowledge of the chapter "King, Rook, and Pawn vs. King and Pawn" can come in handy in diverse situations. For example, consider:
8/3p4/8/8/8/8/3P2k1/5KR1 w
White to Move and Win
A wrong start would be 1. d4?? d5!, since now the position is one of the 6 of 40 drawn positions discussed in "King, Rook, and Pawn vs. King and Pawn." Even 1. d3?? would merely draw. Instead, the key, strangely enough, is to leave the White pawn right where it is. Here is a sample line: 1. Ke1+ Kf2 2. Kd1 Ke2 3. Rg3 Ke1 4. Rf3 Ke2 5. Re3 Ke1 6. Re2 Kf1 7. Kc1 Kg1 8. Rf2 Kh1 9. Rg2 d6 10. Kd1 d5 11. Kc1 d4 12. Kd1 d3 13. Rf2 Kg1 14. Rh2 Kf1 15. Rg2 Ke1 16. Rf2 Ke2 17. Rf4 Ke1 18. Rf3 Ke2 19. Rxd3## [1-0].
Finally, consider:
8/3p4/2p5/6k1/6K1/2PP1R2/8/8 w
White to Move and Win
Here, White can't simply take a pawn with his rook, because then the remaining two-on-one pawn ending is drawn. This is because White won't be able to queen without trading Black's last pawn. Instead, the subtle winning technique is to separate kings, trap the Black king, and make his pawns move forward. Let's see a sample line: 1. Rf7 Kf5 2. Kf4 Ke5 3. Kf5 Kf4 4. Kf6 Kf5 5. Ke7+ Kg6 6. Ke8 Kg5 7. Rf6 Kg4 8. Rf5 Kg3 9. Rf4 Kg2 10. Rf3 Kh2 11. Rg3 Kh1 12. Rg2 d6 13. Kf8 c5 14. Kg8 d5 15. Kh8 c4 16. Kg8 cxd3 17. Kh8 d4 18. c4 d3 19. Rd2 [1-0]