1A bank owned and run by the government of a country, especially a central or national bank which issues currency, holds and manages exchange reserves, etc.
2US A bank holding a charter conferred by the state in which it is located, and subject to the banking laws of that state. Also more generally: a bank originating or located in a (specified) state.
Early 18th century; earliest use found in John Humfrey (bap. 1621, d. 1719), clergyman and ejected minister. From state + bank.