A mortgage contract that either states to allow (or not prohibit) a credit worthy buyer from assuming the mortgage contract of the seller. When a buyer assumes the loan of the seller, they are in essence taking over the seller's mortgage and will be making their mortgage payments.

Assuming a loan will save the buyer money if the rate on the existing loan is below the current market rate. Closing rates can also be avoided as well.

"Due-on-sale" mortgages state that the mortgage must be repaid by the sale of the property, and are therefore not assumable.

Bankapedia's Take: 

Most Mortgages nowadays are no longer assumable, however understand those that are will still require that you qualify. If you are simply taking over someones payment this is not assuming the mortgage.