To meet up with someone again, or to get back together, is to reunite. When your wandering cat comes home after a week-long adventure, you can happily reunite with her.
If a couple breaks up and then reconciles, they reunite. And when a large family — all of the aunts, uncles, cousins, and second cousins — has a reunion, they also reunite. Both reunion and reunite come from the Latin root unus, or "one," and the "again" prefix re-, so reunite means "make as one again."