Polyurethane, shellac, varnish and lacquer: use a cotton swab with acetone and apply it to the wood. If it gets sticky, then it's shellac or varnish and if it forms balls, it has a polyurethane finish. If it is lacquer, the lacquer will completely dissolve. Apply a few drops of denatured alcohol to the furniture, as shown in the photo above.
Wait a few seconds and then touch the area with a soft-bristled brush or cloth. Shellac, a popular finish before 1920, will soften and become a little sticky. If not, it's not shellac, so move on to the next test. Wax can usually be identified by feeling the surface.
For lack of a better term, the surface will feel waxy. With the side of a coin, you can lightly scrape off an inconspicuous area and you'll see a buildup of wax on the side of the coin. Add a few drops of mineral alcohol to the surface and cover it with a shot glass. If wax is present, it will become cloudy and should be soft enough to be cleaned with a cloth.
A more aggressive test would be to use a few drops of ammonia under a shot glass. This will cause the wax to turn yellow and rise; HOWEVER, ammonia will also damage many other finishes. This should only be done in an inconspicuous area or if you intend to remove all the finishing layers. Use a clean cotton swab moistened with acetone and apply it to an inconspicuous place.
If the acetone forms pellets, it is most likely a polyurethane-coated wood. If not, look at the stain over the course of a minute or two. The lacquers and varnishes will become sticky, while the lacquer will completely dissolve. Finally, to differentiate between shellac and varnish, use a clean cotton swab to apply denatured alcohol.
The varnish will react slowly, but the lacquer will dissolve immediately. Once you've discovered the finish of your furniture, a good general guideline is not to use wax products in oil finishes and polyurethane finishes, and not to use drying oils in finishes that don't contain oil.