The Problem

There are three big 'Uh Oh' ingredients commonly found in nail polish: formaldehyde- a known carcinogen, dibutyl phthalate- a suspected reproductive toxin, and toluene- known for skin and respiratory irritation. In addition, to remove the polish you need to use a remover containing a new chemical cocktail.
What Can I Use Instead?
A variety of companies have begun to formulate polish without these three chemicals. Avoiding these chemicals doesn't mean you have to skimp on colour options or quality- I recommend Cheeky Monkey polish, which is formulated without formaldehyde, toluene, dibutyl phthalate, camphor and parabens. Natural removers can be difficult to find, but I recommend Suncoat's Natural Nail Polish Remover Gel.
I suggest taking a look at the Skin Deep Cosmetics Database to find more products and options to make your manicure worry-free.