Something that's rightful is deserved or legitimate. If you accidentally grab someone else's coat when you leave school, you'll have to bring it back and return it to its rightful owner.
You'll frequently find rightful describing the word owner: "Police were quickly able to return the puppy to its rightful owner after the hoodlum grabbed it from the back yard." Anything or anyone with a legal right to something is considered rightful, so a newly elected U.S. president takes their rightful place as leader of the country as soon as they're sworn in.