We often hear that the Virgin Mary was “an unwed Mother,” because while She was still unmarried (only “betrothed”), the angel Gabriel announced that She was pregnant. It's an inspiring story for unwed mothers, but it happens to be not true.
Mary and Joseph weren’t simply “betrothed,” in the sense we imagine that term (as fiancées). They were already married. That’s why Matthew 1:19 refers to Joseph as Her “husband,” and says he contemplated “divorce.” Needless to say, if they weren’t married, neither of those details would make sense.
The source of the confusion is that traditional Jewish weddings were in two stages; first, the couple would wed, and then, the husband would then have a certain amount of time to prepare a place for his new wife. Jesus uses this to describe His own relationship to His Bride, the Church, in John 14:2. He goes before us (to Heaven) to prepare a place for His Wife.
That's the short answer to the question, "Was Mary an Unwed Mother?" For a longer, more detailed answer, explaining the traditional Jewish wedding practices in more detail, check out this post.