Buyers, sellers, real estate brokers, even bankers get confused, sometimes, by the difference between "mortgagor" and "mortgagee".
The confusion arises because they're mixed up about who actually does the mortgaging. Most homebuyers say they'll go to a lender "to get a mortgage" and real estate brokers do the same: "We'll find a bank to give you a mortgage".
That's all wrong, though. You're the one who will give a mortgage to the bank.
A mortgage is a financial claim against your property. You sign a document giving that claim (a lien) to the lender, and in return they give you -- money.
You give the lending institution two things: first a note or bond, which is a personal promise to repay the loan, just like any promissory note. Then you also sign another document, a mortgage (in some states, a trust deed), which says "and if I don't pay as promised, you can take the property."
It's you who mortgages the property. The lender takes the mortgage and holds it (until the debt is paid off.)
So you don't look for a bank to give you a mortgage. You do the giving. You're looking for a bank that will take your mortgage.
The person who performs the action is the "or" or "er" actor. Think of "employer" or "donor". That makes the you, the borrower, the "mortgagor". You're doing the mortgaging.
The bank, or other lender, takes your mortgage. And the recepient is always the "ee" figure. That makes the bank the "mortgagee".
Real estate students remember the difference by noticing that "borrower" has two "o"s in it, and so does "mortgagor." "Lender", on the other hand, has two "e"s, and so does "mortgagee".
Hope that clears things up!