What does Luke 2:2 mean?

Luke strives for accuracy and clarity in his writing. That does not eliminate all possible questions about what he means, or how to interpret it. Here, he establishes the time frame of Jesus' birth (Luke 2:1). The prior verse referred to the reign of a particular emperor. Augustus ruled from 27 BC to AD 14.

Here, Luke's comment about Quirinius and his governorship of Syria creates a puzzle. So far as we know from history, Quirinius held his position around AD 6 or 7. That would be several years after the actual birth of Jesus. As commonly translated into English, this verse indicates Mary and Joseph are participating in the "first registration" under that ruler. However, the Greek word prōtos is translated elsewhere as "before," as in John 15:18 and Matthew 27:64. Some translators suggest this might be better rendered to indicate this was a registration "before" Quirinius was governor.

Others note that a decree and a registration, themselves, are two separate events. This suggests that the decree went out, and people went to their ancestral hometowns to be recorded, but the formal "registration" itself did not happen until later, when Quirinius was governing. That might have been a delay partly caused by the death of Herod (Matthew 2:19). Still others believe, based on secular history, that Quirinius may have held his position twice.

Time and history take their toll on details, and this is one example. It's not possible to establish a single, perfect, clear answer for Luke's meaning here. Ultimately, it makes no difference; the various explanations are reasonable enough that this is simply a question of "what" Luke intended, not whether the events occurred.