A courtship is a period in a romantic couple's relationship when they are dating. Most partners go through a courtship before deciding to get married.
Courtship is an old-fashioned word, assuming that two people who love each other will eventually get married. If your uncle and aunt only met a few weeks before their wedding, you can say they had a brief courtship — and if you have friends who aren't married but have been together for years, you could describe their decades-long courtship. The word is indeed old-fashioned, from the 16th century when it meant "paying court to a woman with intention of marriage."