Money transfer licenses in the United States are covered in full under Money Transmitter License (MTL), which explains who needs one, how states regulate the activity and what applicants must provide. "Money transfer license" is an everyday name for the same thing, used especially by people outside the industry and outside the United States.
In US law the regulated activity is called money transmission, not money transfer, and licensing is mainly a state matter. There is no single federal license. A business that receives money to send to another person or place generally needs a license from each state where it serves customers, and almost every state has such a law, Montana being the long-standing exception. Many states process applications through the Nationwide Multistate Licensing System, and applicants typically must show minimum net worth, post a surety bond, pass background checks and run an anti-money-laundering programme. A model law, the Money Transmission Modernization Act, aims to make state rules more consistent, and it has been adopted in a growing number of states.
Separately, most money transmitters must register with FinCEN as a money services business. That federal step is a registration, not a license, and does not replace state licensing. Other countries use different names, such as payment institution or remittance licences, and their rules differ.