To move a piece, simply specify the piece's designation letter, and the square you wish to move the piece to. If two or more of the same type of piece can move to the square in question, you must specify either the row or column of the piece you wish to move.
Pawns are a bit different.
For a simple pawn move, you just specify the destination square. For example e4 moves the King's pawn to the e4 square. To capture a piece with a pawn, you specify the column the pawn is in, an 'x' and the destination square. I.e. exd5 would have the pawn on e4 capture the piece on d5. The notation is the same for an en-passent capture in which exd5 would signify the pawn on e5 capturing the pawn on d5, but the pawn would still move to d6.