On a pawn's first move from its starting square, it may be moved either one or two spaces forward, providing there are no other pieces occupying either the square it reaches or the one it passes over.
Pawns attack forward on the diagonal, so if it moves two spaces, it may "pass over" a space attacked by another pawn. If it does so, the opponent's pawn may capture it as if it had moved only one square, but only on that following move, never later. This is called a capture en passant or "in passing." The attacking pawn takes the pawn's pace on the first square, not the second.
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A pawn on the 2nd rank may move two spaces directly forward if the way is clear. Pawns do not jump.