I'm going to repeat the warning about the carpet backing. Do NOT pour polish remover, acetone, etc. on your carpet. It will soak into the backing, and then the glue that holds the yarns to the backing will dissolve. You suddenly have a bare spot where the yarns come out.
Apply whatever you are using to an old rag first, and apply the solvent to the carpet using a blotting motion. Do not rub or over-agitate as you will remove the nail polish, but have a fuzzy (ugly) spot instead. Our goal is to restore the carpet so it looks nice.
Lee's secret ingredient? Lighter fluid. Don't smoke around it (true of any of the cleaners mentioned so far). Lighter fluid is cheap, and it's volatile so it dries fast. Blot completely when done. A rinse with a mild detergent solution afterwards can help diminish any odor of any of these solvent based cleaners.
Otherwise, everyone has provided you with good advice.