What does Luke 2:1-2 tell us about the time of our Lord's birth? I know they were ordered to enroll, but I do not know what this means.
What this means is that Joseph and his family were required to register themselves as residents of the Roman Empire and their province. Joseph was required, according to this chapter of Luke, to go to Bethlehem to be counted because that was the ancestral home of his family. There is some dispute, though, as to the accuracy of these verses.
In the verses that you cite, we are told that Jesus was born at a time when Augustus was emperor of Rome and Quirinius was governor of Syria. The census was, according to Luke 2:1, ordered by Augustus for the entire Roman Empire. The problem with this is that there was no such census ordered at the right time. Augustus did order censuses of Roman citizens at least three times in his reign, but Joseph was not a Roman citizen and would not have needed to be counted.
Quirinius did order a census some years after Jesus is believed to have been born. This census would have included Joseph and his family. However, Luke 2:1 explicitly says that Augustus ordered the census. Furthermore, Luke 1:5 says that Jesus was conceived at a time when Herod was ruling, not when Quirinius was. These things cause problems for us in understanding the historical circumstances of Jesus’ birth.
What is clear, though, is the meaning of Joseph going to be enrolled. He was going to have his name and that of his family put on the census rolls in his ancestral home town.