“It cannot be too greatly emphasized that the most important role in pawn endings is played by the king.” Siegbert Tarrasch
King and pawns endgames are of vital importance for a few reasons because they can introduce you to some key concepts of endgame theory.
Endings in which the stronger side has a pawn and the weaker does not can be split in two categories:
The pawn can queen without help
While these ending can be calculated manually every time, the rule of the square is a safe and quick way to answer the question “can this pawn queen?”
The pawn cannot queen without help
When a pawn needs king support then concepts like opposition and key squares are of vital importance.