For a fairly detailed explanation of the Trinity, see this answer. But the gist of it is that God is three in person and one in nature.
Since then, there are three persons in God, they exist in relationship. The terms Father and Son express the relationship between two of the persons; these terms do not speak in any way to the nature of God.
The question of how the relationship of Father and Son makes sense in the divine eternity is somewhat complex. Begetting is a concept which we are less in tune with in our modern western culture, but it speaks to an idea of bringing into existence by the process of reproduction, giving rise to; bringing about. It is the act of reproducing something of the same nature as the original being.
In our experience, begetting is temporally limited; a Father must be born and then mature before he begets. But when he begets it's another person of like nature (note that a person begets of similar nature, but God, being inifinite, necessarily begets of perfectly identical nature).
However, in God's eternal present, begetting occurs in the immediate eternal such that there never was a time when the Son was not, but yet the Son is a second person in eternity arising from the nature of God.
In my thinking, and this is just my thought, it's as if the nature of God necessitates from all eternity that there be three persons for relationship and communion.
As something of an aside, a further point of confusion that arises is because Jesus "put off" his divine nature in order to be incarnate as a man:
6 Who, being in very nature God,
did not consider equality with God something to be used to his own advantage;
7 rather, he made himself nothing
by taking the very nature of a servant,
being made in human likeness.
8 And being found in appearance as a man,
he humbled himself
by becoming obedient to death—
even death on a cross!
9 But we do see Jesus, who was made lower than the angels for a little while, now crowned with glory and honor because he suffered death, so that by the grace of God he might taste death for everyone.
In order to properly be able to redress the disobedience of Adam, it was necessary for Christ to enter our dimensions of time and space and to function in obedience as a man. To this end, scripture teaches he emptied himself of his divine nature; he was the same person, but with a temporarily (that is, within the dimensions of our temporal existence) diminished nature. He chose not to exercise his God-nature except via the Holy Spirit as a man.
So while on earth, Jesus functioned in a limited capacity, as a man invested with the Holy Spirit. Thus he said, "the Son can do nothing by himself; he can do only what he sees his Father doing":
16 So, because Jesus was doing these things on the Sabbath, the Jewish leaders began to persecute him. 17 In his defense Jesus said to them, “My Father is always at his work to this very day, and I too am working.” 18 For this reason they tried all the more to kill him; not only was he breaking the Sabbath, but he was even calling God his own Father, making himself equal with God.
19 Jesus gave them this answer: “Very truly I tell you, the Son can do nothing by himself; he can do only what he sees his Father doing, because whatever the Father does the Son also does. 20 For the Father loves the Son and shows him all he does. Yes, and he will show him even greater works than these, so that you will be amazed. 21 For just as the Father raises the dead and gives them life, even so the Son gives life to whom he is pleased to give it. 22 Moreover, the Father judges no one, but has entrusted all judgment to the Son, 23 that all may honor the Son just as they honor the Father. Whoever does not honor the Son does not honor the Father, who sent him.
Note there is a duality of sorts here; Jesus is still God in person, but has emptied himself of his divine attributes in order to become man.