How does acetone free nail polish remover work? What replaces the acetone but still is effective.?

Acetone is replaced with ethyl acetate (another non-polar organic solvent). Acetone has been known to be toxic so that is why non-acetone remover is popular. It doesn't work as well (as quickly, really) but it does the job. Acetone also evaporates more quickly because it has a VERY low boiling point.