What makes nailpolish melt styrofoam?What is nail polish?

First, it's nail polish remover, which is acetone, that does the trick.

Very simply styrofoam is made up of long strands of polymers with air trapped inside. Acetone is a solvent, which breaks down these strands and lets the air escape. As the air escapes, the styrofoam "melts" down into the small amount of residual polymer.