The opposition is the single most important idea in king-and-pawn endgames.
When the kings stand on the same file (or rank) with exactly one square between them, the player who does not have to move holds the opposition. The opponent must step aside, letting your king advance.
To promote a pawn you usually need your king to reach the squares in front of it. Gaining the opposition forces the defending king to give way, so your king can outflank it and escort the pawn home.
The same idea works with three or five squares between the kings (distant opposition) and on diagonals. The rule of thumb: with an odd number of squares between the kings and it's the opponent's move, you have the opposition.
With an a- or h-pawn the defending king can get stuck in the corner and the position is often only a draw, even a pawn up — opposition can't save it.