No chemical reaction takes place in the removal of fingernail polish. Fingernail polish remover, commonly acetone-based, dissolves the ingredients of the hardened nail polish back into the liquid form you originally applied it to the nail in.
Using the simple chemistry rule “like dissolves like”, the solvent molecules in the remover get in between the polymer chains in the hardened nail polish and push them apart. In addition, notice that once the remover evaporates from the applicant (cotton ball, tissue, etc.), you are left with hardened nail polish. This polish can again be broken down and could, theoretically, be applied and removed numerous times!