Thrifts offer savings accounts and other services as well as mortgages. Mortgage bankers do nothing but lend money. Mortgage brokers are not bankers and have no money of their own to lend. They act as middlemen, working with a number of lenders to find a loan to match a particular borrowers needs. By state law, brokers work for the borrower.
Mortgage brokers work with many lenders including commercial banks, thrifts, and mortgage bankers.
Brokers may also have access to lenders who don't have an office located in your state, but are licensed to lend money there. But the mortgage broker has to be paid. It shouldn't because the broker is processing the paperwork, so it cost less for the lender to make the loan. In return, the lender discounts the loan to the broker. By state law, both the broker's fee and the discount the broker is getting from the lender must be disclosed up front to the borrower.