So if a Relative of mine died and he didn’t finish paying off the house however his spouse still owns the house can a relative take over and pay the remaining mortgage for them.

Anyone can pay the mortgage so long as there is still at least one living mortgagor: the bank doesn't care where the money comes from. So if your grandmother was also on the mortgage, you can pay it for her. You can also move in, if she will let you, and live there. To protect your interests, you probably want to have a contract with her that you will be allowed to live there so long as you pay the mortgage, taxes, expenses, etc. and that once the mortgage is paid in full, she will quitclaim the home to you: speak to a real estate attorney, and he or she can structure the transaction for you and draw up the paperwork--he or she may come to a different suggestion than made above (the law generally provides more than one way to get something done), but the important point is, so long as your grandmother was also on the mortgage, you can do what you want. Just make sure you have written agreements to protect your rights, since oral ones will not do so.
If your grandfather was the only one on the mortgage, then it came due in full when he passed away: the bank has the right to insist on payment in full. If you can pay the balance in full, you can do this and again come to an agreement with your grandmother that she will then sell or quitclaim to you (get a signed written agreement before you pay anything).
If you can't pay the balance, you would likely have buy the house and finance it in your own name, but the bank will probably let you have time to do this (i.e. not immediately foreclose, which would be their right) if you stay in contact with them and work with them on this: they'd rather get the money by having the loan paid off then take the house and have to deal with preserving, marketing, selling, etc. it.