Controlling a pawns critical square is what K+p vs K endgames are all about...
With a pawn on the 2nd, 3rd, or 4th rank the critical squares are two ranks ahead of the pawn, whereas pawns on the 5th rank have their critical squares only one rank ahead of the pawn. That's because of the lack of maneuvering room the opponent's King has... as the superior side advances his pawn and King.
In the diagram White's c5-pawn's critical squares are d6,c6,and b6. If White's King can occupy any of these three squares, he can force a win, no matter what Black does. If Black can prevent White's King from getting to any of these critical squares while the pawn is on the 5th rank, then Black draws.
Thus, with Black to move he draws by 1...Kc7. But if White moves first, he occupies d6, proceeding his pawn to the 6th rank. This is what most K+p vs K endgames are all about; getting the stronger sides King to the 6th rank in front of the pawn.
Once you accomplish that, you win whether the pawn (if not a Rook pawn) occupies the 2nd, 3rd, 4th, or 5th ranks.
With Rook pawns...
The critical squares in this position are a1,b1 and c1. If the White King can occupy any of these squares he will draw.
However, Black to play wins by 1...Kb2 taking the opposition and taking control of the three critical squares and not allowing White's King into the corner.
Occupation of a critical square means the pawn can promote by force, even against the best defense.
White to move draws by 1.Kc1 and the Black King never gains the critical square and the pawn is stopped.
1.Kc1 Ka2 2.Kc2 a4 3.Kc1 a3 4.Kc2 Ka1 5.Kc1 a2 6.Kc2 Stalemate.
1.Kc1 a4 2.Kb1 a3 3.Ka1 a2 ...Stalemate