Palimony is money that a court can order one person to pay, regularly, to a former partner after they split up, even though they were never married. They had been living together as a couple. The word is mainly American, and it sounds like newspaper-and-court language. You will see it in stories about famous people. You will not hear it much in ordinary chat.

Though actually, it is not a right that every unmarried couple has. It depends on the place, and on the story: did they promise to look after each other? did they share money as if they were married? Many countries, including Taiwan, do not use this word in court at all. People here still meet it in US news. It was coined as a kind of blend-joke on alimony, the older word for money paid after a marriage ends.