Incarnation is a word that so many people don't fully grasp. Of course we can't fully comprehend the incarnation, but we can apprehend it in Scripture. The reason we rejoice at Christmas is that the Baby born to Mary and Joseph on that very first Christmas was not an ordinary Child. As Matthew records, this Baby was the ultimate fulfillment of Isaiah's prophecy of Immanuel, or God with us, the ultimate self-revelation of God to humankind. Jesus, the Christ, was, and eternally is, God incarnate or God in flesh.
Although John's Gospel doesn't include a narrative of the birth and infancy of Jesus, the doctrine of the incarnation is aptly summed up in John's introduction to His Gospel where He says "In the beginning was the Word, and the Word was with God, and the Word was God." Then, says John "the Word became flesh and dwelt among us and we beheld His glory, the glory as of the only begotten of the Father, full of grace and truth."
So the clear testimony of Scripture is that in the incarnation Jesus Christ was fully God and fully man; that is, Jesus Christ existed as the perfect unity in one Person of a divine and a human nature, neither co-mingling those natures nor becoming two persons. It is this miraculous incarnation of God that you and I, along with Christians worldwide, celebrate this Christmas.