An American option is an option contract that can be exercised at any point throughout the life of the contract at the discretion of the option buyer. This is in contrast to European options which can only be exercised at expiration. All other things being equal, an American option should demand a greater premium than a European option because it gives the buyer greater flexibility.
The Black-Scholes formula only calculates a fair options price for European options. Prices for American options can be calculated using the Bjerksund-Stensland American Option Approximation model. In the real world, though, Black-Scholes is commonly used to price both European and American options as most American options are not exercised early and therefore behave similar to European options.
Most traded options on individual stocks are American options.