What does it mean that Jesus is God’s only begotten son?
The Bible does not call Jesus the “only begotten Son.” Not exactly. The Greek word which is translated in the KJV as “only begotten” is the word monogenes. The word monogenes means unique one or the one and only one. It is hard to precisely define what God meant when he had John describe Jesus as the monogenes Son, who is himself God. Theologians for almost two thousand years have tried to define in what sense Jesus is the unique, one-and-only Son. The early church fathers talked about Jesus as “begotten, not made.” Again, the meaning of the word begotten in this phrase is a bit obscure. They meant that Jesus was sent by the Father, but not created by the Father. Father is sender and Son is sent one. This seems like a reasonable description of the relationship between Father and Son, but it is more of a theological interpretation than a thing directly said in the Bible. The word “begotten” is really not in John 1:14 or 1:18, so calling Jesus the “only begotten Son” in these passages from the King James translation is a bit speculative. Begotten is an archaic word. For this reason, using it in the modern context to describe Jesus is probably not all that helpful.
Let me assume that I am answering a different question than the one you actually asked, because I assume you are asking what the Bible (not the KJV) says in John chapter 1. “What does it mean in John 1:14 and 1:18 when it calls Jesus the monogenes Son?” My answer is that Jesus of Nazareth, who is the Word of God who became flesh is the Son of the Father in a unique and in a one and only sense. He is God and at the same moment he is Son of the Father who also is God.