Usually, you may apply for a marriage license at any county clerk's office in the state where you want to be married. (In some circumstances, you must apply in the county or town where you intend to be married -- this depends on state law.) You'll probably have to pay a small fee for your license, and you may also have to wait a few days before it is issued.
In some states, even after you get your license you'll have to wait a short period of time -- one to three days -- before you tie the knot. In special circumstances, this waiting period can be waived. If you wait too long, your license will expire. Licenses are good for 30 days to one year, depending on the state. If your license expires before you get married, you can apply for a new one.
For more specific information about marriage license laws in your state, see Chart: State Marriage License and Blood Test Requirements.