Quirinius was governor of Syria: He held this position from AD 6 to AD 9. Earlier, from 6 to 4 BC, he also had authority in Judea, perhaps through a form of joint rule. He may have begun the census (an official count of people for taxes).
Historical records mention three other censuses at that time in Syria, Gaul, and Spain. It would not be unusual to have one in Judea, even though no non-biblical record mentions it.
Jesus was born around 6 to 4 BC, before the death of Herod the Great in 4 BC (see Matthew 2:1–19).