According to Micah 5:2 and Jewish tradition, the Messiah (the Christ) would be born in Bethlehem, a small town near Jerusalem. Luke 2 informs us that Mary and Joseph traveled to the town for a census prior to Jesus's birth, thus fulfilling the prophecy. Some critics, however, see the lack of corroborating historical accounts as evidence that the census and Quirinius's governorship at the time must have been fictitious—a myth invented later.
If this story had been a myth, though, it would have been a rather poor one. First of all, a census at the bidding of the Roman Empire repulsed Jewish sensibilities and desire for sovereignty. This hardly seems the place to begin a legend.
Beyond this, Luke's precise language emphasizes a particular census, as if to contrast it with similar ones. In fact, another census did occur ten years later, which Luke refers to in Acts 5:37. The author's additional information concerning Qurinius's governorship (Luke 2:2), which is unnecessary for the narrative, reveals a familiarity with the recent past. Luke knew his audience would need clarification between similar events, so he gave them the details necessary to understand the date he meant.
The emperor at the time of Jesus's birth, Caesar Augustus, kept count of the population throughout his empire for taxation purposes. Israel would have been no exception. Even if we have no other accounts of the census taken during that time (which is no proof that the event didn't happen given the sparse records available), the event seems likely from what we do know of the Roman Empire.
Quirinius may, in fact, have governed Syria at the time and also ten years later. However, the original Greek suggests another possible reading. Luke's statement may imply that King Herod performed a Jewish style census (counted according to the historic location of the tribes and clans) to keep the peace. Thus, the command of Caesar was not really carried out in the Roman method (counted by where the person was born) until ten years later when Qurinius was governor and Herod had died.
Adapted from The Life and Times of Jesus the Messiah by Alfred Edersheim (Book II, Chapter VI).