No one has seen God at any time; the only begotten God who is in the bosom of the Father, He has explained Him.
John 1:18 NASB
For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life.
John 3:16 NASB
Begotten means that Jesus is the unique one and only son, not created, eternally existent with the Father. With the word, “begotten,” the Nicene Creed was countering Arius’ error that Jesus was at one time created, “There was a time when the Son was not.” Begotten does mean uniqueness, but also is a way of saying that Jesus, God’s Son, was not created. Jesus was eternally existent with the Father, there was not a before and after “begotten” moment. These quotes from Wayne Grudem, Fred Sanders, and B.B. Warfield explain:
As for the texts that say that Christ was God’s “only begotten Son,” the early church felt so strongly the force of many other texts showing that Christ was fully and completely God, that it concluded that, whatever “only begotten” meant, it did not mean “created.” Therefore the Nicene Creed in 325 affirmed that Christ was “begotten, not made”:
‘We believe in one God, the Father Almighty, Maker of all things visible and invisible.And in one Lord Jesus Christ, the Son of God, begotten of the Father, the only-begotten; that is, of the essence of the Father, God of God, Light of Light, very God of very God, begotten, not made, being of one substance with the Father….’
This same phrase was reaffirmed at the Council of Constantinople in 381. In addition, the phrase “before all ages” was added after “begotten of the Father,” to show that this “begetting” was eternal. It never began to happen, but is something that has been eternally true of the relationship between the Father and the Son. However, the nature of that “begetting” has never been defined very clearly, other than to say that it has to do with the relationship between the Father and the Son, and that in some sense the Father has eternally had a primacy in that relationship.
Wayne Grudem, Systematic Theology: An Introduction to Biblical Doctrine, pg. 244 (original edition).
If we call it eternal begetting or eternal generation, we are only guarding ourselves against possible misunderstandings. It is not that once upon a time the Father begat the Son, having previously not begotten the Son. No, the eternal Father and the eternal Son have always existed together, the Son always standing in this relationship of from-ness or begottenness from the Father.”
Fred Sanders, The Deep Things of God: How the Trinity Changes Everything (pp. 91-92). Good News Publishers/Crossway Books. Kindle Edition.
Warfield notes that “only begotten” stands without the article, which points up the idea of quality rather than individuality. He reminds us further that “only begotten” does not convey the idea of subordination or derivation but of uniqueness and consubstantiality: “Jesus is all that God is, and He alone is this.”
Fred G. Zaspel, The Theology of B. B. Warfield: A Systematic Summary (Kindle Locations 5460-5463). Crossway.