Nail polish or lacquer is a coating applied to the nails to protect the nails and to remove this polish, an organic solvent is used which is called Nail Polish Remover.
The base solvent in nail polish remover is acetone or ethyl acetate. In the past Acetonitrile was used, however after two cases were reported of poisoning of young children, acetonitrile has been banned in cosmetics including nail polish removers.

How does the Organic Solvent work?

The nail polish remover is the organic solvent that is used as an ingredient in the nail polish. The hardened nail polish is dissolved by the nail polish remover, basically putting it back into the form that it was in when applied to a nail. For dissolution to occur, no chemical reaction takes place. However, what is necessary is that the solvent likes the polymer. There is a rule of thumb that may be a bit oversimplified, but it works, and that is "like dissolves like." And in this case, it can really be oversimplified and said that the organic polymer that is the hardened polish on a nail can be dissolved by an organic solvent. So if a solvent likes the polymer, as is the case for nail polish and nail polish remover, the solvent molecules which may be ethyl acetate or acetone, get in between the polymer chains and "push" them apart.