Floorboard rot is usually cause by moisture. Once a floorboard has sustained significant rot damage, there's no way to salvage it, and it will have to be replaced. The replacement process is complicated by the tongue-and-groove edging of the boards, which will prevent you from pulling the bad boards straight up and out. See your flooring dealer for replacement boards of the same size and style as the bad ones.
Set your circular saw blade depth to equal the thickness of the rotted floorboard (based on the thickness of the replacement board). It will generally be around ¾ inch.
Run the circular saw up the length of the centre of the rotted board. Don't cut into the sides of the board.
Cut additional lines up the middle of the board, running them side by side and then crossing them over each other, so chunks of the wood start coming out. Continue until you've got as much out as possible without cutting into the sides of the board.
Knock the edges of the board into the cut-out middle, using your hammer and chisel. Don't damage the tongue and groove milling on the adjacent boards. Pull out any nails you come across, using the hammer.
Set the replacement floorboard on the work bench, face down. Use a razor knife to cut off the bottom side of the groove that runs along one long edge of the board.
Set the replacement board into the space where the rotted board was. Set it in first on the tongue side, pressing it into the groove of the adjacent board, then drop the other side down. The missing bottom half of the groove will allow it to sit down over the tongue of the adjacent board.
Shoot finishing nails though the surface of the new board, every 10 to 12 inches along both edges, using your nail gun.
Repeat for each rotted board as needed.