While in most states if operating your Scooter or Moped on private property you do not need a motorcycle license. However, if you plan to use one on a public road you may be required to have a license. As with all vehicles you should also check with your local DMV to see if you need to be titled and registered as well.
Here is a break down of the states and the license required:
States that DO require a motorcycle license to ride a Scooter or Moped:
Alabama, Alaska, Arkansas, Connecticut, Delaware, Georgia, Hawaii, Idaho, Maryland, Minnesota, Montana, Texas
States that DO require a motorcycle license if your Scooter or Moped can exceed 25mph:
Arizona, Colorado, Kansas, Massachusetts, Michigan, Mississippi, Nebraska, Nevada, New Hampshire, New Jersey, North Carolina, Ohio, Pennsylvania, Rhode Island, South Carolina, South Dakota, Utah, Vermont, Virginia, Washington, West Virginia, Washington DC
States that DO NOT require a motorcycle license to ride a Scooter or Moped (You only require a normal drivers license):
Florida, Illinois, Iowa, Maine, Missouri, New Mexico, New York, North Dakota, Oklahoma, Oregon, Wisconsin, Wyoming
States that Require a specialized license to ride a Scooter or Moped:
California requires either a Moped or M2 class license
Kentucky requires a Moped specific license
States that DO NOT require any license to ride a Scooter or Moped:
Indiana, Louisiana, Tennessee
Hopefully this list will help you better understand your local licensing requirements for riding your Scooter or Moped on public roads.