King + Pawn vs King

King and pawn versus a lone king is the most basic endgame of all: can you promote that last pawn or not? The answer comes down to the opposition, the key squares, and whether it's a rook-pawn (which is trickier to win). Master this and the rest of endgame play gets much easier.

Source for further study: Dvoretsky, Endgame Manual, ch. 1.