The most common use for a lacquer is as a protective coating. A typical lacquer coating is composed of a resin or a cellulose ester or both that's dissolved in a highly volatile solvent. Solvents used in lacquer include acetone, ethyl acetate and methyl ethyl ketone, or MEK. It's the resin or cellulose ester in lacquer that's the polymer, and those two substances act as binding agents. So lacquer contains polymers but they're thinned out through use of solvents.
A polymer is a natural or synthetic chemical substance made up of many molecules that are strung together in long chains. Resin, which is a secretion of many different types of plants, is composed of long chain molecules that make it a classic natural polymer. Cellulose ester is a synthetic substance made in a laboratory and it, too, is a polymer with many long chain molecules. A polymer, then, is the base substance of a lacquer but it's not a lacquer itself.
Combining a polymer like resin or cellulose ester and a solvent like acetone together will create pure lacquer but other additives give lacquers their specific characteristics. For example, many lacquers contain pigments, plasticizers and antioxidants to give them color and character. In addition, as the solvents in an applied lacquer coating evaporate away the polymers left behind dry and make the lacquer hard and durable. The polymers in a lacquer don't evaporate because they're not volatile, unlike the solvents contained in a lacquer.
Polymers can be coatings all by themselves. A classic polymer coating is Teflon, which has been used to coat everything from cookware to bullets. Lacquer is also often referred to as varnish. Because of the solvents contained within them, it's recommended that you apply lacquers using gloves and a mask or respirator. Lacquer and shellac are cousins, but shellac's main ingredient is "lac." Lac is a resin obtained only from certain insects.