The chess endgame with a king and a pawn versus a king is one of the most important and fundamental endgames, other than the basic checkmates (Lasker 1915). It is important to master this endgame, since most other endgames have the potential of reducing to this type of endgame via exchanges of pieces. It is important to be able to tell quickly whether a given position is a win or a draw, and to know the technique for playing it. The crux of this endgame is whether or not the pawn can be promoted (or queened), so checkmate can be forced.
In the first paragraph of one of his books on endgames, Peter Griffiths emphasized the importance of this endgame:
There is simply no substitute to a clear understanding of when and how these positions are won or drawn, not only so that one can play them accurately, but in order to recognize in advance what the correct result should be. If you can do that, you can exchange off quite confidently from a more complex position (Griffiths 1976:1).
In the positions in which the pawn wins, at most nineteen moves are required to promote the pawn (with optimal play) and at most nine more moves to checkmate, assuming that the pawn was promoted to a queen (Levy & Newborn 1991:144).
Except for the section on defending and some actual games, it will be assumed that White has a king and pawn and Black has a lone king. In general, Black should place his king in the path of the pawn to try to prevent its promotion.
The first thing to realize is that the pawn may be able to queen unassisted by his king, simply by advancing to the queening square before the opposing king can capture or block the pawn. The rule of the square is useful in determining whether the pawn can queen unassisted, or whether the king can stop the pawn. In this position, the pawn is on the fifth square from the queening square (counting the queening square itself). A square of five by five squares with the queening square in one corner and the pawn in an adjacent corner can be imagined. (Often, the easiest method of constructing the square is to draw a diagonal mentally from the pawn to the last rank; this is the diagonal of the square). If the black king can move into this square, he can stop the pawn, otherwise the pawn wins the race. In this position, if it is Black's move, he can move to b4 and enter the square, therefore he can stop the pawn. If it is White's move, the pawn advances, the square shrinks to four by four, and the king cannot move into the square, so the pawn queens (Müller & Lamprecht 2007:15). See for further discussion on the rule of the square.