"Vault cash" is the money that a bank will keep on premises (which is usually kept in their vault) to deal with their day-to-day cash needs.
The bank must have a certain amount of cash on hand in order to deal with these types of every day transactions. "vault cash" is considered to be part of a bank's reserves, and banks will usually be required to have a certain amount of "vault cash" on hand.
Bank reserves are essentially an antidote to panic. The Federal Reserve obliges banks to hold a certain amount of cash in reserve so that they never run short and have to refuse a customer's withdrawal, possibly triggering a bank run.
The required bank reserve follows a formula set by the Federal Reserve Board's regulations that are based on the amount deposited in net transaction accounts. These include demand deposits, automatic transfer accounts, and share draft accounts.