Lacquer and Varnish are names of two materials used as a coating, on surfaces like wood, metal, etc. These words can also be used as verbs meaning to apply that material (like 'to paint' means 'apply paint')
Varnish is a general term for any clear or transparent finish. In the old days it was always made of some gum or resin, like wood resin, and a solvent like turpentine. But today there are many modern materials, including oil-based and water-based varnishes made out of very strong urethane (plastic).
Lacquer is a specific kind of clear varnish, that uses 'lacquer thinner' as a solvent- a mixture of butyl acetate and xylene or toluene. This is a much lighter, thinner, and quicker drying solvent, and also it is much more nasty. Lacquer thinner fumes are quite toxic, and you have to use a respirator mask to work with them. (if you smell nitrocellulose lacquer, you have already gotten brain damage!)
The advantages of lacquer over other finishes are several:
1) because of the solvent, lacquer is very fast drying
2) because of the solvent, wet lacquer dissolves into dry lacquer when itis applied on top. (enamel does not!)
3) lacquer can be hand-polished to a very beautiful, fine shine
4) lacquer is very thin- only a few hundredth of a millimeter. This makes it especially good for musical instruments because it does not interfere with the vibration and resonance of the wood. It is also good for fine furniture or anywhere fine detail is desired.
The big disadvantage of lacquer, aside from the toxicity of the solvent, is that it is relatively delicate. Unlike modern durable urethane varnish, lacquer is easily scratched. When exposed to humidity and temperature changes it tends to crack and become rough. (this can be easily reversed and polished using a solvent) This is why they stopped using lacquer on cars- before about 1960 most cars were painted with lacquer, and the colors faded and peeled terribly. So lacquer is finer but much weaker.