The pawn moves one square forward (never backward). There are two exceptions to this rule.
One, the pawn is the only piece that captures differently than it normally moves. The pawn captures one square diagonally. In the diagram below, if it is white's turn to move, white can capture black's pawn on e4 with its pawn on d5.
Two, the pawn may move one OR two squares forward on its initial move. The white pawn on b7 can move to b6 or b5 on its initial move.

If a pawn reaches the final rank (horizontal row), it may be "promoted" to another piece. This is done by simply removing the pawn and replacing it with a piece of higher rank. A player may add a second queen, or replace the queen lost earlier in the match. Any other piece may be substituted as well, however the queen is usually the piece of choice.