The Bible never deals with the doctrine of the Trinity as an abstract truth, but reveals the trinitarian life in its various relations as a living reality, to a certain extent in connection with the works of creation and providence, but particularly in relation to the work of redemption.
— Berkhof, L. (1938). Systematic theology (p. 85). Grand Rapids, MI: Wm. B. Eerdmans publishing co.

As we listen to God reveal Himself in Scripture, we hear three persons speaking of themselves as one God.  

The Father, the Son, and the Spirit each posses all the fullness of the Divine Essence.  All that makes God God abides fully in the Father, the Son, and the Spirit.  They are not three parts of the Godhead, nor are they three members of some Divine species, like three members of the human race.  There are not three Gods, only one.

The Father begets the Son in an an eternally, complete, and necessary personal relationship of Sonship.  That is to say, there never was a time when the Son was not, nor was there a time when the Son was only partially begotten or in the process of becoming.  He was always there as the Son dwelling in the bosom of His Father.  We speak of this begetting as necessary because the Father could not be the Father without the Son, nor could the Son be the Son without the Father.

As we discuss this process, we must also take care to protect the dignity of each member of the Trinity.  Although the Son is begotten from the Father and is, therefore, Son because of the Father, He is not God because of the Father.  The Son is autotheos - God in Himself, of Himself, and from Himself.  The same must be said of the  Spirit as He proceeds from both the Father and the Son.  He is the Spirit of the Father and the Son because of this personal relationship, but He does not become God because of this relationship.  He, too, possesses all of the fullness of the Godhead in, from, and of Himself.

This is why, as we listen to God speak, we hear 3 persons:  each existing the way only God can be, to each are ascribed Titles only God should hear, each doing the works only God could do, each receiving the worship only God should get, from each come down gifts only God could give, and each knowing, loving, and guarding the Christian the way only God could do.

If you are a believer in the Lord Jesus Christ, take comfort:  This great Triune God has never existed without loving you in the Son.  The Father has always looked at you with Christ colored spectacles.  The Son has carried away the infinite burden of your Sins.  The Spirit will not stop applying to you the fruit of this redemptive love.  Once you are His, you are His forever.