How do I fix a sunken floorboard?
Looks like either there's no subfloor under that board, or the subfloor failed (rotted?) in that area. If this is ground floor, it might be worth looking at this area up from the basement or crawl space or whatever's under there, to see what was supposed to be supporting that board and what condition it's in. 

That's assuming that there was something there. Someone may have done a sloppy install over a hole and simply hoped that nobody would ever put weight on it.

If you can get to it from underneath, and it was never properly supported, I'd be inclined to see if I could push it back up into position, then nail some support in place under it -- one or two 2x4's between the joists, tight up against the floor, then insert shims to fill any remaining gap between those and this board. Not very elegant, but more elegant than leaving a board floating in midair.

If you can get to it and discover that it was originally supported but Something Has Gone Wrong, you'll have to proceed from there.