We don't know.

The most commonly used sources are the gospels of Matthew and Luke, the two books that contain nativity narratives. But they agree on very little, and it's difficult to reconcile either of them with any outside evidence.

Both Matthew and Luke mention that Jesus was born during the reign of Herod the Great. Herod is believed to have died in 4 BCE, so that would seem to put Jesus' birth before then. Luke also states that Jesus was about 30 years old when his ministry began, and that this was in the 15th year of Tiberius' reign as emperor. Tiberius' reign began in 14 CE, so that means Jesus' ministry began in 28-29 CE. If "about 30" is taken to mean "in his early 30s," a date of birth between 6 and 4 BCE is plausible. The census that took place under Quirinius (said by Luke to be the reason Mary and Joseph were in Bethlehem) occurred in 6 CE, though this is considered to be a less reliable piece of information because it's mentioned only by Luke, while both Luke and Matthew put the date during the reign of Herod. John agrees that Jesus' ministry began in 28-29 by putting it in the 46th yeast of the construction of the Temple in Jerusalem.

The month and day are even less clear. The traditional date of December 25 was chosen centuries later. No reliable source gives information about the time of year. Some modern biblical literalists claim that the shepherds being out in the fields in Luke's narrative suggests a date in September or October. It's also possible to work from the narrative in Luke 1, describing the conception and birth of John the Baptist, and connect that with the conception of Jesus. This method would also give a birth date in early autumn. Of course, these estimates depend on the reliability of Luke 1 and 2, which many modern scholars doubt.