The king and pawn vs king endgame is one of the most fundamental endings in chess: one side has a king and a single pawn against a lone king. The outcome — win or draw — depends entirely on the relative positions of the two kings and the file the pawn is on.
The central concept is opposition: two kings are "in opposition" when they face each other with exactly one square between them on the same file or rank, and it is the side not having the move that holds it. The side with the pawn typically needs to seize the opposition to escort its pawn to the eighth rank and promote. The rule of the square lets you quickly determine whether the defending king can catch the pawn: mentally draw a square with one side running from the pawn to the promotion rank; if the opposing king is inside that square (or can step into it on its turn), it catches the pawn.
One critical exception: when the pawn is a rook pawn (a- or h-file), the endgame is a theoretical draw if the defending king manages to reach the corner, as the attacking king cannot dislodge it without causing stalemate.
In practice, before simplifying into this endgame, always check three things: is the opposing king outside the pawn’s square (and unable to enter it)? Can your king seize the opposition? Is your pawn a rook pawn that risks ending in a draw? These three questions should guide your decision on whether to trade into this ending.