A state chartered bank offers commercial and retail bank services. It's main role is to accept and safeguard customer deposits and to lend money out. Basically, it's like any other bank besides it's funds come from the state itself and the bank can be one that only exists in that state.
For example; A charter school(virtual school, homeschooling) is basically the same thing in the form of a school; it is paid for by the state, those who live in the same state that pays for the school get to attend