The main job of the Bank Branch Manager is to manage the everyday operations of the bank. For example they may look over loan applications for certain applicant, while also supervising tellers and personal banks and interviewing and hiring job candidates.
Branch Managers also handle customer issues/problems, report to the regional manager and discipline employees if their branches policies are broken. From time to time they will also open accounts (checking, savings, cds, IRAs etc.).
That's just an overview of what Bank Branch Managers do, but it's a good way to begin understanding what authority it provides to them.
In terms of customer issues the bank Branch Manager can act as the liason to solve issues. For example, they may approve the removal of overdraft fees for a customer if they feel it will best assist the customer and the bank, they may also remove certain ATM fees, online banking fees and other charges if they see fit.
Branch Managers also have the authority in many cases to approve loans set forth by personal bankers. They may examine in more detail a borrowers history and ability to pay and determine if specific loans should be approved or denied.
They may also open accounts for customers including checking, savings, cds, IRAs and other accounts. Typically this is done by a personal banker, however in various cases they may handle larger accounts or assist customers during business hours when wait times are longer.
Essentially the Branch Manager is also the last stop for customer complaints and services, allowing customers to fix issues that personal bankers are not allowed to fix or refuse to fix, while providing lending overrides and denials in certain cases.
Bank Branch Managers also determine who works for them by completing job interviews with qualified (and sometimes not-so qualified) candidates. It is typically up to these managers to determine who they hire to work with them. This process may include running background and credit checks, interviewing and completing all necessary new hire paperwork and assignment of training.
Branch Managers may also need to discipline employees for failing to follow their branches policies, this may include write ups, suspensions and ultimately firing employees for blatant violations of their banks policies.
Finally, bank Branch Managers may be given some authority to provide local donations as part of the banks outreach programs. While outreach is not always offered by every branch, if there is money available for sponsoring baseball teams, donating to charities and for other purposes it is typically the bank's branch manager who will provide these funds and other types of support.
Essentially the bank Branch Manager serves the same role as a manager at most companies, they assist customers with issues, monitor and discipline employees, hire and fire employees and work with local interest groups. Essentially they control the interests of the banks day to day business.