How to win using Lucena position? For the ones that never heard of the Lucena position, it a position where we have a KRP+KR ending, and the side with the extra pawn has his king in front of the pawn.

The problem is that when you try to move the king out of the pawn's way, the enemy rook will deliver annoying checks that will not stop unless you are familiar with the Lucena position and setup your rook in a certain way that will eventually be able to block the checks from the enemy rook.

(example of not knowing how to Lucena)

[fen "1K1k4/1P6/8/8/8/8/r7/2R5 w - - 4 1"]

1. Rd1+ Ke7 2. Kc7 Rc2+ 3. Kb8 Ke6 4. Ka7 Ra2+ 5. Kb6 Rb2+ 6. Kc6 Rc2+ 7. Kb5 Rb2+ 8. Kc6 Rc2+ 9. Kb6 Rb2+ 10. Ka7 Ra2+ 11. Kb8 *


How do I use the Lucena position in my favor?
The Lucena is one of the most important endgame positions to know, and the nickname for it helps a bunch when trying to figure it out over the board. Remember this phrase - "Building a Bridge."

The idea is to get your rook to the fourth rank. Although the move looks rather odd, the idea is simple. Once you bring your king out, he needs to be protected. Your king can advance all the way to the fifth rank, and after the check the rook can block and your pawn can queen. Allow me to use your example to demonstrate the Lucena:

[fen "1K1k4/1P6/8/8/8/8/r7/2R5 w - - 4 1"]

1. Rd1+ Ke7 2. Rd4! Rb2 3. Kc7 (3. Ra4 Kd7 4. Ka7 Kc7 5. b8=Q+ Rxb8 6. Rc4+ {works as well, as referenced by @dfan in the comments}) Rc2+ 4. Kb6 Rb2+ 5. Kc6 Rc2+ (5... Rb1 6. Rd5! Rc1+ 7. Rc5 Rb1 8. Rb5) 6. Kb5! Rb2+ 7. Rb4


And the pawn queens easily. The very first check is important because it forces the opposing king an extra file away from the queening square. As you can see, in the final position the black king is one tempo too slow to stop the pawn from reaching the eighth rank.