Assumption Clause

You want to sell your house, which still has some payments left to go on the mortgage. You have found a buyer. The usual procedure is for the new buyer to get their own mortgage for the property, which provides the money to buy the house. You take that money to pay off your mortgage.
The key point to notice is that there are two mortgages involved in this process. The buyer got a mortgage and used that mortgage to pay off your mortgage.
If your mortgage has an assumption clause, it can simplify this process. The assumption clause allows you to pass the mortgage off to someone else. Basically, the buyer takes over your mortgage, rather than getting another mortgage.
Most mortgages don't have assumption clauses. They are nice for the person buying the house, because the buyer can then effectively choose between the old mortgage and the terms they can get from a lender now. So if the interest rate of the assumable mortgage is lower than the interest rate they could get today, the buyer might choose to assume the mortgage. However, if rates are lower now, they can scrap the old mortgage and just go the traditional route in getting a new one.
As you might guess, this fact has made assumption clauses unpopular with lenders, thus assuring that they have become a rare inclusion in mortgages today. However, a similar process does come up in mortgages backed by the government's Federal Housing Authority, which allows mortgage assumption under certain conditions.