There are a few different terms for getting married without legally registering the marriage with the government. The most common term is probably common law marriage, although this term has different meanings in different states. In general, though, common law marriage refers to a relationship between two people who live together and consider themselves married even though they haven't gone through a formal ceremony or gotten a marriage license.
There are also marriages that happen through religious ceremonies but not legal ones. These might be called spiritual marriages or faith-based marriages. Again, the rules around these kind of marriages vary from state to state, so it's important to do some research if you're considering this option.
Finally, there are also couples who simply never get around to getting legally married. This isn't really an intentional choice; it just happens sometimes when life gets busy. Couples in this situation are often referred to as unmarried spouses or domestic partners.
So there's no single answer to the question of what it's called when you get married but not legally. It depends on the specific situation and context. But whatever you call it, know that there are lots of other couples out there in similar situations!