What Does "Son of God" Mean? How is Jesus the "Son of God"?

There is a pernicious error present in our world amongst those outside of the Church, who claim that Jesus Christ is the "Son of God" and that means He is not God Himself.  This blasphemous error attacks the very core of what we profess: belief in the Holy Trinity. 

Is Jesus Christ truly and really God?  Yes, He is.  But is He also the "Son of God"?  Yes, He is.  Then what do we mean when we say He is the Son of God if He is God?  Let's explore this now so as to refute those who are in error.

The Catholic Encyclopedia entry on "Son of God" serves as a good basis:

The title "son of God" is frequent in the Old Testament. The word "son" was employed among the Semites to signify not only filiation, but other close connexion or intimate relationship...The title "son of God" was applied in the Old Testament to persons having any special relationship with God. Angels, just and pious men, the descendants of Seth, were called "sons of God" (Job 1:6; 2:1; Psalm 89:7; Wisdom 2:13; etc.). In a similar manner it was given to Israelites (Deuteronomy 14:50); and of Israel, as a nation, we read: "And thou shalt say to him: Thus saith the Lord: Israel is my son, my firstborn. I have said to thee: Let my son go, that he may serve me" (Exodus 4:22 sq.).

The title "the Son of God" is frequently applied to Jesus Christ in the Gospels and Epistles. In the latter it is everywhere employed as a short formula for expressing His Divinity (Sanday); and this usage throws light on the meaning to be attached to it in many passages of the Gospels. The angel announced: "He shall be great, and shall be called the Son of the most High... the Holy which shall be born of thee shall be called the Son of God" (Luke 1:32, 35). Nathaniel, at his first meeting, called Him the Son of God (John 1:49). The devils called Him by the same name, the Jews ironically, and the Apostles after He quelled the storm. In all these cases its meaning was equivalent to the Messias, at least. But much more is implied in the confession of St. Peter, the testimony of the Father, and the words of Jesus Christ.

And so we see the title "Son of God" is not meant to refer to filiation in that Jesus is the Son of a Father and not equal to the Father in the way that I might be called the son of my earthly father.  Rather, His title of "Son of God" is rather meant to express His Divinity and show His inseparable connection with His Heavenly Father, the 1st Person of the Blessed Trinity.

So the next time you hear someone erroneous claim that Jesus Christ is not God since He is the "Son of God," all you need to do is refute the errors using Scripture as your guide.  Remember, the errors of Arianism which claim that Jesus is not God are still very much around yet to this day.  For good measure, be sure you understand the basic Theology of the Holy Trinity.