No one knows the real birthday of Jesus! No date is given in the Bible, so why do we celebrate it on the 25th December? The early Christians certainly had many arguments as to when it should be celebrated! Also, the birth of Jesus probably didn't happen in the year 1 but slightly earlier, somewhere between 2 BCE/BC and 7 BCE/BC, possibly in 4 BCE/BC (there isn't a 0 - the years go from 1 BC/BCE to 1!).
There are many different traditions and theories as to why Christmas is celebrated on December 25th.
A very early Christian tradition said that the day when Mary was told that she would have a very special baby, Jesus (called the Annunciation) was on March 25th - and it's still celebrated today on the 25th March. Nine months after the 25th March is the 25th December! March 25th was also the day some early Christians thought the world had been made, and also the day that Jesus died on when he was an adult. The date of March 25th was chosen because people had calculated that was the day on which Jesus died as an adult (the 14th of Nisan in the Jewish calendar) and they thought that Jesus was born and had died on the same day of the year.
Some people also think that December 25th might have also been chosen because the Winter Solstice and the ancient pagan Roman midwinter festivals called 'Saturnalia' and 'Dies Natalis Solis Invicti' took place in December around this date - so it was a time when people already celebrated things.
There's a strong and practical reason why Jesus might not have been born in the winter, but in the spring or the autumn! It can get very cold in the winter and it's unlikely that the shepherds would have been keeping sheep out on the hills (as those hills can get quite a lot of snow sometimes!).
During the spring (in March or April) there's a Jewish festival called 'Passover'. This festival remembers when the Jews had escaped from slavery in Egypt about 1500 years before Jesus was born. Lots of lambs would have been needed during the Passover Festival, to be sacrificed in the Temple in Jerusalem. Jews from all over the Roman Empire traveled to Jerusalem for the Passover Festival, so it would have been a good time for the Romans to take a census. Mary and Joseph went to Bethlehem for the census (Bethlehem is about six miles from Jerusalem).
In the autumn (in September or October) there's the Jewish festival of 'Sukkot' or 'The Feast of Tabernacles'. It's the festival that's mentioned the most times in the Bible! It is when Jewish people remember that they depended on God for all they had after they had escaped from Egypt and spent 40 years in the desert. It also celebrates the end of the harvest. During the festival, Jews live outside in temporary shelters (the word 'tabernacle' come from a latin word meaning 'booth' or 'hut').
Many people who have studied the Bible, think that Sukkot would be a likely time for the birth of Jesus as it might fit with the description of there being 'no room in the inn'. It also would have been a good time to take the Roman Census as many Jews went to Jerusalem for the festival and they would have brought their own tents/shelters with them! (It wouldn't have been practical for Joseph and Mary to carry their own shelter as Mary was pregnant.)
The possibilities for the Star of Bethlehem seems to point either spring or autumn.