Thanks for your question.
The short answer is that nobody knows exactly. The year that seems to have the most votes in favour of it is 6 BC. In any case the birth of the Lord must have been at least during or before 4 BC because that is the year of the death of Herod The Great.
Here is a timeline taken from Wikipedia the online Encyclopedia:
c. 6 BC - Suggested birth (earliest)
c. 5 BC - Herod the Great's death (earliest)
c. 1 BC - Herod the Great's death (latest)
c. 6 AD - Suggested birth (latest), Quirinius census
c. 26/27 AD - Pontius Pilate appointed governor of Judea
c. 27 AD - Suggested death (earliest)
c. 36 AD - Suggested death (latest)
c. 36/37 AD - Pilate removed from office
The consensus favours a date between 4 BC and 6 BC as the date of our Lord's birth. However, the exact date isn't really of that much importance. The fact of his birth, his life and work, his sacrifice and his resurrection are what really matter. The fact that he will soon return to fulfill his prayer, "Thy Kingdom come; Thy will be done in earth as it is in heaven" is of primary importance to us while speculation about the exact year of his birth, while of interest, is, relatively speaking, of little importance.
Was Jesus born in late December? Probably not. "The climate of Palestine...the cold of the night, from December to February, is very piercing, and it was NOT the custom for the shepherds of Judea to watch their flocks in the open fields later than about the end of October." -(Alexander Hislop, The Two Babylons, 1959, p. 91).
A far more likely scenario is that Jesus was born in the early autumn, around the time of the Feast of Tabernacles (Leviticus 23:34-36), when Joseph and Mary would have traveled to Jerusalem to keep the Feast along with thousands of other Jewish families. This also helps us understand why in Bethlehem, a few miles to the south of Jerusalem, "there was no room for them in the inn" (Luke 2:7). The town would have been crowded with other travelers keeping the Feast at this time of year and moreso given that a Roman census had also been ordered. What days/month does the week-long Feast of Tabernacles fall on? It varies. Here are the dates for the feast of Tabernacles in recent years:
Jewish Year 5766 : sunset October 17, 2005 - sunset October 24, 2005
Jewish Year 5767 : sunset October 6, 2006 - sunset October 13, 2006
Jewish Year 5768 : sunset September 26, 2007 - sunset October 3, 2007
Jewish Year 5769 : sunset October 13, 2008 - sunset October 20, 2008
Jewish Year 5770 : sunset October 2, 2009 - sunset October 9, 2009
"December 25th was the day the Romans celebrated the birth of their (pagan) sun-god." -Encyclopedia Americana, vol.6, pg 623
"To facilitate the acceptance of the faith by the pagan masses, the Church of Rome found it convenient to institute the 25th of December as the feast of the birth of Christ...from the pagan feast, celebrated on the same day in honor of the pagan god, Mithras..." (Manual of Liturgical History, 1955, Vol. 2, p. 67).
The celebration of Christmas is viewed by many as the participation in an essentially pagan festival made blasphemous by inserting Christ into the place of honour previously reserved for pagan gods.
The bottom line is that nobody really knows on what day or in what month or even the exact year Jesus was born. In terms of real significance, it doesn't matter. December 25 was chosen by the Catholic church to celebrate the birth of Christ in order to co-opt the Pagan festivals attending the end of the old year and the ushering in of the new especially the pagan feast of "Sol Invictus" celebrated on what is now called Christmas Day. It was not chosen because it was the actual birth day of the Lord.
It is important to understand that Jesus' birthday is not a matter of celebration in the New Testament. His death, burial, and ressurection is celebrated, usually weekly (although the frequency doesn't matter), in what most churches call the "communion" service and in the rite of adult baptism by full immersion in water (see Romans 6).
I hope you have found this helpful.