A type of financing in which one person may take over the mortgage from another. For example, Buyer 1 wants to buy a house, so he takes out a mortgage (borrows money from the bank to pay for the house). If Buyer 1 wants to sell the house to Buyer 2 before the mortgage is paid off, and the loan is an assumable mortgage, Buyer 2 may "step into the shoes" of Buyer 1 and take over the mortgage.
If this were not an assumable mortgage, Buyer 2 would have to go to the bank, get his own loan, and use that to buy the house off of Buyer 1. Buyer 1 would then use the proceeds to pay off his mortgage.