Acetone is a solvent in nail polish remover that can be very drying to the nail and surrounding skin and cuticle. It can cause the nails to become brittle.
Removers free of acetone use fruit derivative solvents that don't create the dryness. Especially for fake nails, acetone-free remover is recommended because the acetone can actually dissolve the glue!
Acetone-based removers have the advantage of working faster, especially on deeper, darker colors, but that is not a good enough reason to harm your nails.
Acetone removes polish faster than non-acetone but it is not as healthy, especially for someone who already has dry nails. Look for a remover with a moisturizer and acetone free for healthier removal and only use remover once to twice a week.