So let's start analyzing the concept of key squares in the simple ending king and pawn versus king. I will be cribbing a bit from Chapter 1 of Secrets of Pawn Endings by Karsten Müller and Frank Lamprecht. If you are rated above 1200 and you haven't read this book, you really should. If you are rated above 1200 and if you are serious about improving your game, and you think you don't need this book, I would like to gently suggest that you are woefully mistaken.
If you're a beginner, then Pandolfini's Endgame Course is a good place to start. The endings we'll be talking about are covered in Chapter 5 of that book.
Black's king seems perfectly placed for defense. The pawn on e7 has only one square it can move to, the promotion square e8. If there were Black pieces on the board to capture, the White pawn could dream of becoming a queen on d8 or f8. But a solo king can't be captured.
Let's suppose it's White to move. If White makes a waiting move that doesn't protect the e7 pawn (say 1.Ke5), then Black will just play 1...Kxe7. Unfortunately for White, the only waiting move White has that protects the e-pawn is 1.Ke6. But that move also delivers stalemate. The perfectly placed Black king can't take the protected pawn, its escape squares on the seventh rank, d7 and f7, are both attacked by the White king, and the d8 and f8 squares are attacked by the pawn on e7. Black has to move, but Black is not in check and has no legal move. That's stalemate, and that's a draw. So our first conclusion: if it's White's move, the game is drawn.
But if it's Black's move, Black has only one legal move: 1...Kd7. Black is on the perfect defensive square, but cannot stay there. Black does not want to make a move, but the rules of the game compel Black to do so. That predicament is the essence of zugzwang: "I don't want to move; I must move." 1...Kd7 is answered by 2.Kf7. The White king protects the queening square e8, and Black has no time or means to drive the White king away. The pawn will promote on the next move. Our second conclusion: if it's Black's move, White wins.
Incidentally, Position 1 is particularly interesting because it's the most basic example of reciprocal zugzwang: neither player wants to be the player on move.
We can begin to make a few tentative conclusions from Position 1. The promotion square e8 is a key square: if White controls that square and if Black has no way of capturing the d-pawn, White will definitely win with best play. But if Black occupies or controls the White's pawn's queening square, that may not be sufficient to draw, as zugzwang may compel Black to abandon the square.
POSITION 2 (After Müller & Lamprecht 1.02)
White wins, no matter whose move it is; the Black king can be anywhere on the board.
In Position 2, White controls the promotion square e8 from f7. Black has no way to drive the White king off of f7 using zugzwang, as White can always choose to push the pawn. So the f7 square is a key square for king and d-pawn.
In fact, all of the six squares highlighed (d7, e7, f7, d8, f8, and the promotion square e8) are key squares for king and e6-pawn versus king. The only exception we have to make to this general rule is the obvious exception that White can't allow Black to take the pawn:
POSITION 3 (A silly and hopefully obvious point)
It does no good to occupy a key square if you let your defender capture your last pawn! But if Black can't capture the White pawn immediately, and the White king is on any one of the six key squares, then White will win.
Enough for today. Next time, we'll talk about these two famous positions and the concept of the opposition. Winning the opposition is only a tool that players use to fight for key squares. Do you know at a glance the important difference between these two positions?
Try to figure out these two positions on your own, and we'll talk about them next time.