The answer to that question is not exactly clear. We know the precise date of Christ's crucifixion, but we do not know the precise date of His birth. We know that he was born in 4 BC, however. We know that because the monk who started keeping time relative to Christ's birth made a mistake in his calculations. This threw the calendar off by 4 years. Thus, even though "BC" means "before Christ," Christ was actually born in 4 BC.
When in 4 BC was He born? We do not know. We know it was NOT in December, however, because the Scriptural evidence clearly rules that out. December 25th was originally a pagan holiday, worshipping the birthday of the sun. Christian missionaries tried to Christianize that holiday by calling it the birthday of the Son (of God), and it stuck.
Biblical scholars agree that Christ was born sometime between April 1 and September 31, 4 BC. I think that April is the most likely month, because of these 4 arguments:
1) The fact that the shepherds were "keeping watch over their flock by night" (Luke 2:8), would seem to indicate, according to some historians, that it was "the spring lambing season, when nights would have been balmy and shepherds would have needed to be awake, tending the ewes." The spring lambing season is mostly in April. (J. I. Packer, Merrill C. Tenny and William White, Jr., The World of the New Testament (Nashville: Thomas Nelson Publishers, 1982), p. 115.)
2) In 243 AD, the church produced an official feast calendar called De Pascha Computus, which placed the date of Christ's birth on April 2. It is a historical fact that many in the early church celebrated Christ's birth in April (Paul Lee Tan, Encyclopedia of 7,700 Illustrations (Rockville: Assurance Publishers, 1979), n. 2714. )
3) There are historians such as Edward Reese and Frank Klassen who wrote The Reese Chronological Bible (Minneapolis: Bethany House Publishers, 1980), who state that the origin of April Fools Day is in Christ's Birth. Early Christians were often derided by the Jews and Greeks, and these historians claim that since Christ's Birth was on April 1, the Christians were called "April Fools."
4) Another argument comes from the idea that Jesus is the Lamb of God. In keeping with that idea, it would make sense that He was born at the season when other passover lambs were born (John 1:29). The passover lamb was to be "a male of the first year" (Exodus 12:5), which means that it was born the previous April. (Edward Reese and Frank Klassen, The Reese Chronological Bible (Minneapolis: Bethany House Publishers, 1980), Introduction and p. 1250.)