A mortgage broker provides loans to clients using relationships with mortgage bankers or lenders. They usually have many different sources to fund loans.
A mortgage banker funds loans from lines of credit that they have established with banks. They then package and sell their loans on the secondary market.
A mortgage lender is an entity who makes a real estate loan. ie. mortgage banker, a bank, an insurance company, etc...
A broker or banker can lock your rate and have your loan underwritten. A mortgage banker would hold this on their line of credit until they sell it. Then the servicer (the one who receives your payments) holds the loan for the owner of the mortgage back security. The servicer could still be the mortgage banker that provided your mortgage, but it could also be a different servicer.
Sallie Mae is affliated with student loans. I think you might have been referring to Fannie Mae (FNMA) or Freddie Mac (FHLMC), these conforming loans are underwritten to guidelines published by these entities. This allows mortgage bankers to streamline the sale of loans on the secondary market. The problems that have been disucssed in the media about non-conforming loans.
Selling loans on the secondary market allows the lender to lend out there money, many times over instead of once every 30 years.