In this endgame, you (white) want to promote your pawn. If you promote to a queen or to a rook, you can win the game. One way to think about your task is to see the file your pawn will travel on as a hallway. The black king is like a bully in the hallway. The white king is you. The white pawn is your younger brother or sister. Or, if you don't have a sibling (or don't like your sibling), the white pawn is your pet. The white rabbit in the photo is my pet Abba. Isn't he cute in his hallway?
Do not put your sibling or pet in front of you as a shield. Instead, you must bravely face the black king yourself. In the diagram below, as you sat on e3, you made a scary face at the bully on e5. At least it probably was, because he must either back up (to e6) or step out of the hallway (to f5, d5, f6, or d6). Actually, the name for your scary face is the opposition. You have it, he doesn't. But he's tough. He doesn't want to let you and your sibling (pet) through, so he stays in the e-file hallway with 1…Ke6 The black king is in the hallway. In the next sequence of moves, the white king uses the opposition to either drive the black king out of the hallway or back further in the hallway.
2. Ke4 Kd6 3. Kf5 Ke7 4. Ke5 Ke8 5. Ke6 Kf8 Now that the black king is out of the hallway, and is at the end of the board (8th rank), the sibling/pet/pawn can advance. The enemy may step in the hallway again briefly, but then we will use opposition to force him back out again. Therefore, we have reached a new stage in this endgame. Time for another diagram to show what happens once we have the king out of the hallway and at the end of the board.
1. e4 Ke8 2. e5 Kd8 3. Kf7 Controls the promotion square of e8. 3….Kd7 4. e6+ Kd8 5. e7+ Kd7 6. e8(Q)+ One explanation of how to checkmate with a king and queen against king can be found in Bounding Owl’s article. One method is to keep the queen a knight’s move away from the king, to restrict its movements until it is driven to a side of the board. At that point, bring in the white king to checkmate.
1….Kc7 2. Qe6 Kb7 3. Qd6 Ka7 4. Qb4 Now the black king is restricted to the a-file. 4….Ka6 5. Ke7 Ka7 6. Kd7 Ka8 7. Kc7 Ka7 8. Qa5# 1-0.
To summarize, in this ending you are a brave king with a pawn, and that pawn is as dear to you as a sibling or a pet. Therefore, you first clear the file for your pawn and protect him from the enemy king (the dreaded bully). In the second stage, after you have ejected the enemy king from the far end of your pawn's file (hallway), you may advance your pawn. The third and final stage is to promote your pawn and checkmate.
Time for a quiz to see if you remember the first step to forcing an enemy king out of a hallway. Here is your quiz position.