If the king is on the sixth and the pawn is on the fifth (and it's not a rook pawn), then it's a win for the side with the pawn, no matter who has the move:
[FEN "4k3/8/4K3/4P3/8/8/8/8 w - - 0 1"]
1. Kd6 Kd8 2. e6 Ke8 3. e7 Kf7 4. Kd7 Kf6 5. e8=Q *
With Black to move:
[FEN "4k3/8/4K3/4P3/8/8/8/8 b - - 0 1"]
1... Kd8 2. Kf7 Kd7 3. e6+ Kd8 4. e7+ Kd7 5. e8=Q+ *
If you back it up, though, the position is a win with Black to move and a draw with White to move. If White has to move first, Black can simply mirror the movements of the White king until White moves the pawn forward.
[FEN "8/3k4/8/3K4/3P4/8/8/8 w - - 0 1"]
If the pawn is in front of the king, the general strategy is to make sure to oppose the enemy king when it goes alongside its pawn, to prevent it from getting in front.
[FEN "4k3/8/4P3/4K3/8/8/8/8 w - - 0 1"]
1. Kd6 Kd8 2. Ke5 Ke7 3. Kf5 Ke8 (3... Kf8?? {This loses! If you take this critical square too soon, before the enemy king is alongside the pawn, you can't stay on the square when the enemy king does advance.} 4. Kf6 Ke8 5. e7 Kd7 6. Kf7) 4. Kf6 Kf8 5. e7+ Ke8 6. Ke6 {Stalemate.} 1/2-1/2