A true Mortgage Banker is a lender that is large enough to originate loans and create pools of loans which they sell directly to Fannie Mae, Freddie Mac, Ginnie Mae, jumbo loan investors, and others. Any company that does this is considered to be a mortgage banker. They can vary greatly in size. Some may service the loans they originate, but not all of them will. Most true mortgage bankers have wholesale lending divisions.
Examples of two of the largest mortgage bankers are Countrywide Home Loans and Wells Fargo Mortgage. One is associated with a bank and the other is not, but both are most correctly classified as mortgage bankers.
A lot of companies call themselves mortgage bankers and some deserve the title. For others, it is mostly marketing.