Rub the surface of the dresser with denatured alcohol on a door edge or other hidden surface to determine what kind of stain is on your dresser. Shellac comes right off the dresser, water-based polyurethane softens, and oil-based varnish does not soften at all. Purchase a chemical stripper appropriate for the stain.
Dip your paintbrush into the chemical stripper, and brush it onto the dresser. Let it sit for 10 to 15 minutes according to the manufacturer's instructions.
Scrape the wood with a putty knife to remove the varnish. Allow the excess varnish to dry.
Sand the surface with 80-grit sandpaper, then move down to 220-grit for finer sanding.
Vacuum the dresser surface to clean the wood.
Apply shellac to the surface of the wood using a cotton rag. Pat the shellac onto the wood surface.
Dip your brush into the wood stain. Brush the stain onto the dresser, running the bristles of the brush along the grain. Allow the stain to dry.
Dilute 1 part gloss with 2 parts thinner, and wipe onto the stain.
Apply a top coat of the stain. Apply two more coats the diluted gloss, letting it dry between layers.