Cut the damaged plank with a jig saw or saber saw so that the damaged portion can be removed from the deck. Use a single cut across the damaged board so that the board is split into only two pieces. Take care not to damage the adjacent deck boards.
Pry the damaged portion of deck board loose from the underlying joists by placing a piece of scrap wood on the deck and using a crowbar to pry the damaged board up, starting from one end and gradually moving to the other. Remove the nails or screws fastening the damaged board to the joists as you go.
Mark the placement of the nearest joist under the damaged portion of board decking on the board, then lift the board up and use the square to mark the center-line of the joist across the face of the board.
Move to the other end of the damaged board and mark the placement of the nearest joist past the damaged area. Again, lift the board up and use your square to mark the center-line of the joist across the face of the damaged board.
Cut the damaged plank along each of the center joist lines to remove the damaged portion of the board; leave the undamaged portion(s) of the board ready for re-attachment.
Measure the length of the opening for the replacement board and mark the replacement board accordingly. Then cut the replacement board to match the opening.
Place the replacement board into the opening on the deck and secure it to each underlying joist using nails or screws that match what was originally used to build the deck. You may want to drill pilot holes for each nail or screw prior to attachment. Be sure to counter-sink screws to make them flush with the surface of the deck.
Sand the surface of the installed replacement board, especially at the ends, to ensure a snug, smooth fit. Then treat the board with sealant, stain or paint that matches the original decking.