Damages sections of boards should be cut out and replaced, or a new board fitted if the damage is substantial. First check that there are no gaps or cables running below the damaged section, otherwise you will have to remove the entire board in case you cut into them by accident.
To cut a section of board, first find the edges of the joists at each end. Do this by sliding a knife blade along the gap between the boards. If the boards are tongue-and-grooved, you will have to cut through the tongues by drilling a starting hole and using a pad saw or with a circular saw set to the depth of the board.
Drill a starting hole for the pad saw just in from the edge of each joist and cut through the board at each end in line with the joist edges. Lift out the damaged section; If it is nailed to intermediate joists, lever it free using a bolster and chisel and a stout length of wood. Lever the board upwards at the fittings with a chisel until you have lifted the end enough to be able to slide the wood below it, while resting it on the tops of the boards on each side.
Pushing down on the end of the board will spring the fixings from the joist. Continue in this same fashion until you have freed the board. A complete floorboard can be removed in the same way. Screw or nail lengths of 50mm batten to the sides of the joists flush with the undersides of the old boards. Then nail a new section of floorboards in the tops of the battens.