Qualifications
Each state has its own specific qualifications, but generally, you have to be at least seventeen, earn a high school diploma or a GED, have a certain amount of hours completed at a beauty school, pass the licensing exam, and, finally, pay the fee. After that, you're good to go.
In New York, for example, you need to have taken 250 hours of classes at beauty school and have passed both written and practical exams as a nail technician to get your license (source). That's actually pretty lenient, though—in some states, you need as many as 600 hours of class. The national average is around 375 hours of class completed (source).
You'll also have to pay a licensing fee (about $100, less to renew) (source).
If you open up your own salon, you'll also need a license to operate a business.