Non-Member Banks
  
You know how Rudolph was left out of all the reindeer games? Well, non-member banks are left out of all the banker games.
Eh, okay. The membership here has to do with the Federal Reserve. A bank can be chartered either on a national level or at the state level. Federally chartered banks have to become members of the Federal Reserve, the United States' central bank, the entity reponsible for overseeing the currency and conducting monetary policy. (It also fulfills major regulatory functions for the financial industry.)
State banks don't necessarily have to become members of the Fed. They can. But they don't have too. The ones that don't join up are known as non-member banks.
This group of banks are charted as state institutions, but that doesn't mean they are completely immune to federal regulation. True, the Fed isn't their main regulator. But they still fall under federal jurisdiction. Istead, they get overseen on the national level by the FDIC.
Even with the alternative regulatory structure, these banks aren't completely divorced from the Fed. They can't delete the Fed's contact information and stop sending them Christmas cards. Non-member banks still have to follow certain Fed rules. For instance, they must maintain federally-mandated reserve requirements.
Meanwhile, they aren't completely left out of the Fed-run banker games, either. For example, these non-member banks can still use the Fed's discount window.