Was Jesus really born on the 25th December?
The answer to this question is that no date is given for Jesus birth in the Bible, and no-one knows the true date. Indeed, for the first 300 years of the church there was no celebration of Jesus birth at all!
However, it was eventually decided that it was appropriate to celebrate the remarkable event of God coming to earth as a man, and a date had to be chosen. The Christian church has always taken the existing culture around it and reshaped it with Jesus at the centre, and this is what they did now.
The 25th December already hosted two other festivals (natalis solis invicti (the Roman "birth of the unconquered sun") and the birthday of Mithras, the Iranian "Sun of Righteousness" whose worship was popular with Roman soldiers). Also the winter solstice, another celebration of the sun, fell just a few days earlier. Church leaders decided to commandeer this date and give it a new and much more important meaning – the birth of Jesus Christ.
Christmas was first celebrated in 336AD and today, the older festivals have faded away and many Christians think of the 25th December as the date when God entered our world as a baby. The important thing about this date is not that it was the actual day when Jesus was born, but that this is the day when we remember that God came down to live with us.
(As a matter of interest, some Christian traditions celebrate Christmas on the 7th January rather than the 25th December. There is no problem with this, because the date is not an actual birthday but rather a time for us to remember.)