You can fix many defects in the surface of a solid or hollow-core wood door with wood filler, sandpaper and glue, but depending on the level of damage, gluing on a new skin may be a better option. A door skin is simply a sheet of 1/8-inch wood veneer, and attaching it to the door is similar to attaching wood veneer to a tabletop to any other flat surface. You need a strong adhesive, and although you may find a good water-based one, many are petroleum-based and require plenty of ventilation.
Pop the hinge pins with a screwdriver and hammer, and take the door down. Lay it flat on a pair of sawhorses and remove all the hardware, including the hinges, knob, deadbolt and any metal hangers or trim.
Degloss existing finish or paint by washing the door with a sponge and a solution of 1/2 cup trisodium phosphate to 1 gallon warm water. Let the door dry, then sand it with a pad sander and 120-grit sandpaper.
Pull off any loose veneer or cut it off with a utility knife. Fill spaces left by missing veneer with wood filler, troweling the filler on with a putty knife and sanding it flat when it dries. If you're re-skinning both sides of the door, prepare the other side the same way.
Prepare a sheet of veneer to fit on the door by cutting it to size with a utility knife. Make it about 1 inch wider and 1 inch longer than the door. Lay it on top of the door, arrange it so the wood grain runs perpendicular to the door's long edge, and mark the two upper corners of the door on the veneer with a pencil.
Remove the veneer and lay it flat on a work bench or on a sheet of plywood laid across a second pair of sawhorses, with the marks facing up. Don a respirator. Apply a layer of contact cement to the veneer and to the surface of the door with a 4-inch paint roller. Let the cement dry until it is tacky, which takes about 10 minutes.
Turn the veneer over, align the two marks with the two top corners of the door, and lay it down. You may need someone to help you do this. When the veneer is in place, press it down with a 4-foot piece of a two-by-four. Hold the wood on edge and draw it along the wood grain, working from the center of the door toward the ends.
Turn the door over and score the veneer along one edge of the door with a sharp utility knife. Make two or three more passes to cut through the veneer. Trim the other three sides of the door in the same way.
Sand the edges of the veneer by hand with 150-grit sandpaper to smooth it out and remove any splinters. Sand the entire surface of the door by hand in the same way before applying finish to the veneer.