Description: A counting of peoples across the Roman Empire, requiring that all people return to the lands of their origin. In Joseph's case, that was Bethlehem, the city of David.
Significance: This not only places Jesus as very much part of the world around him, as opposed to a mythical figure living in some kind of secondary, ethereal otherworld, it also links him directly into the prophecies about where he would come from. This is crucial, as it is a prophecy that, were he a mere mortal man, he would be unable to control.
First sighting: The census only emerges specifically once in Luke.
"In those days Caesar Augustus issued a decree that a census should be taken of the entire Roman world. (This was the first census that took place while Quirinius was governor of Syria.) And everyone went to their own town to register."
(Luke 2, Verses 1-3)
Explanation: There has been some considerable debate about the census, because there are what appear to be discrepancies with other sources for the known history of the period. Specifically, the Bible seems to indicate that Jesus was born during the reign of Herod the Great.
Now we know that the calendar dating to exactly the time of Jesus's birth is difficult, which is why the year 0AD is more a best guess. Therefore, it isn't in itself surprising to learn that Herod died in 4BC. However, the census that historians have recorded from that period, mentioned in the writings of the Jewish historian Josephus, says that the census of governor Quirinius took place in 6AD. So what is happening here?
Luke could have meant a number of things all of which would have been accurate. In other words it would have been clear to him and to those reading his record at the time what he was meaning.
That is because the translation of the original language is less than certain on this point. In the Bible, there is a footnote to the verse where Luke says "took place while Quirinius was governor of Syria". But the translation could also mean that it took place "before" Quirinius was governor of Syria. This would make it perfectly plausible that Jesus was born during the reign of Herod the Great, eliminating the contradiction.
Secondly, the word "governor" is in question here. Luke doesn't actually use the normal Greek word for "governor" in this case, but actually uses the more common word for "hegemon" or "overlord". This could mean that Quirinius wasn't in fact the official governor yet, but was in fact a significant figure in the local Roman authority who had a great deal of influence. This is further confirmed by the writings of Justin Martyr in the second century, who listed Quirinius as a "protector" rather than giving him the official title of governor. Therefore, it's quite possible that the exact times here are less clear, and many more times are possible.
Thirdly, it isn't as though Rome only had one census that came round every so often. There were tax censuses, designed to give an idea of exactly how much money the government could bring in, but there were also allegiance censuses, where rather than merely counting everyone, people were gathered up and encouraged to swear a pledge of allegiance to Caesar in Rome.
Josephus mentions an occasion "When all good people gave assurance of their good will to Caesar". These types of tributes would also require an enrolment of individuals from across the empire. Orosius, a fifth century Christian, links this registration with the birth of Jesus saying that "all of the peoples of the great nations were to take an oath". Also, although the census recorded in 6AD was one of the biggest, it wasn't the only one (as is made clear when Luke says "first" census). Many others may have happened on a smaller scale to organise other matters. Therefore, it is possible that the widely recorded census of Quirinius is not in fact the census which happened when Jesus was born.
The census is important because it is what connects the accounts not only to the specific time, but also to the specific place. Jesus was to be born in Bethlehem, and it is the census that seemed to cause Joseph to take his family there.
What is not entirely clear from the passage is whether or not the Romans required people to return to their homelands or not. Much has been made of this by people debating the accuracy of the gospels, with many arguing that it is a completely absurd suggestion that you would need to travel back home for a census. However, the passage doesn't say that it was the law. Joseph may have had reasons of his own for going back there, such as having relatives there who could help care for Mary, or that was where records of his tax status had been kept.
It would appear that there was a legal need for the census respondents to return to their own home, seeing as how the inns in Bethlehem were full, and Mary would have come with Joseph because the duteous role of the husband was far more strictly enforced. Mary was his wife, and therefore he could not in good conscience leave her in the care of another, even if she was heavily pregnant. Later records indicate that the practice of homeland census records did exist. In 104 AD, there was the taxation edict of Egypt, under prefect Vibius Maximus, which said "The enrolment by household being at hand, it is necessary to notify all who for any cause whatsoever are away from their administrative divisions to return home in order to comply with the customary ordinance of enrolment, and to remain in their own agricultural land".
The evidence then is clear. A census did definitely happen. Exactly what it was for and how it operated is unclear, but there is no reason to doubt that it was there, just as Luke described it.