Bring the wood table to an open, well-ventilated work space or outside. Place a drop cloth under the table. If the table comes apart, break it down into as many components as possible.
Pour varnish thinner onto a clean cloth. Blot the thinner onto the entire surface of the wood and allow it to soak for about 10 minutes. If you notice the thinner starting to dry before the 10 minutes is up, apply more thinner.
Scrape the table with a plastic putty knife or a soft-bristled scrub brush soaked in the varnish thinner. Keep scrubbing and applying more varnish stripper until the finish has been completely removed and you are left with bare wood.
Wipe the wood clean with a solution made with 1/4-cup trisodium phosphate and 1 gallon of water. Rinse the table and allow it to dry completely.
Restain the table with a stain of your choice. You may use a paintbrush, roller, sprayer or a clean cloth to apply the stain. Always apply with the grain. Wipe the wood with a clean, dry cloth until no more stain comes off of the wood. If you are using a pigmented varnish, you may skip this step.
Dilute the varnish with an equal amount of mineral spirits or varnish thinner. Brush the diluted varnish onto the table using full, even, smooth strokes. Overlap each stroke by about 1/4 to 1/2 inch. Allow the varnish to dry overnight.
Scuff the varnish with a piece of 400-grit sandpaper. Wipe away the dust with a damp cloth. Apply two coats of undiluted varnish using the same method used earlier, and allow each coat to dry thoroughly.