An "assumable" loan means you can transfer the mortgage to somebody else. However, there are some restrictions. Lenders want to make sure you transfer the mortgage to somebody who will continue to pay. That person needs to have decent credit and sufficient income.
In general, it is difficult to transfer a mortgage. Your best bet may be if you have an FHA loan or VA loan.
Instead of transferring a mortgage to somebody else, you may be able to refinance the loan. You repay the old loan, and replace it with a brand new one. Again, someone has to qualify for the loan with acceptable credit and income levels.
It may be tempting to transfer a mortgage "unofficially". You can get creative, and you can also get in trouble. Lenders may forbid some of the strategies that come to mind. In addition, you are still on the hook - financially responsible for repaying the loan - unless you do things officially.
How to Move Forward
If you want somebody to assume your mortgage, do it the right way. Talk with your lender to see if you can transfer the mortgage to anybody else, and what the requirements are.
In cases of death, divorce, or "less formal" sales, work with an attorney. The fees you pay may save a lot of headaches and money down the road.