Trinity Sunday, the first Sunday as Ordinary Time resumes after the Easter season, is a day when Christians are called to especially dwell on what is the central great mystery of the Christian faith: the Trinity. There is only one God, but that one God has revealed himself to us as eternally existing as three distinct, co-equal, co-eternal persons, the Father, the Son, and the Holy Spirit.
As this doctrine of the Church most accurately (even if mysteriously) describes the very nature of God, it holds an important place in the Church's Hierarchy of Truths. However, it is also one of the most misunderstood doctrines.

First, what the Holy Trinity is NOT:

Mistake 1) Each Person of the Trinity has a "different job" - always be wary when a teacher or preacher addresses the Trinity as "Creator, Redeemer and Sanctifier". This is an example of an ancient error called modalism: believing that the three persons of the Trinity are just three "modes" or masks that God switches between: sometimes God is in "Father" mode, sometimes he switches to "Son" mode, and other times he’s in "Holy Spirit" mode – but he’s only in one "mode" at a time. The orthodox (the one in line with the Church) teaching is that all three Persons of the Trinity exist eternally and concurrently. The Persons are not just "different jobs" or forms God has.

Mistake 2) It's really three "gods" - this is simply incorrect. In the three Persons of the Trinity, Catholics only believe there is one God. That's why we begin the Creed on Sundays and holydays with “We believe in one God…” The three Persons of the Trinity  - Father, Son and Holy Spirit - are divine, but they are not three Gods. Rather, they share the same divine substance (or, as the Creed says about the Father and Son, they are “consubstantial”).

Mistake 3) The Trinity didn't exist in the Old Testament; it only came about during New Testament times - Although it is true that the Son and the Holy Spirit are revealed most clearly in the New Testament, that doesn’t mean that they weren’t around before that. The three persons of the Trinity, including the Son and the Holy Spirit, exist eternally and act together in perfect harmony. This is hinted at from the very earliest references in Genesis, the first book in the Bible: Then God said: Let us make human beings in our image, after our likeness (Gen. 1:26, emphasis added). The language used is a clear allusion not to an individual, but to a community (and it isn't referring to angels, as humans are created in the image and likeness of God, not angels). Other examples in the Old Testament can be found here.

So, what then IS the Holy Trinity? Unfortunately, most physical analogies that try to describe the spiritual reality of the Trinity fail in one way or another. The best description that we have of this mystery appears to be available is a historical one known as the Athanasian Creed. Named after the great saint of the 4th Century, it is the most clear and thorough way to describe the Trinity in physical terms. It is also quite densely-packed and needs to read through carefully (and probably more than once); even then, be assured that you won't completely understand the Trinity - as a mystery, it may never be completely understood by us limited humans. Here it is:

Whosoever will be saved, before all things it is necessary that he hold the Catholic faith. Which faith except every one do keep whole and undefiled; without doubt he shall perish everlastingly.

And the Catholic faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence.

For there is one Person of the Father; another of the Son; and another of the Holy Ghost. But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one; the Glory equal, the Majesty coeternal. Such as the Father is; such is the Son; and such is the Holy Ghost.

The Father uncreated; the Son uncreated; and the Holy Ghost uncreated. The Father unlimited; the Son unlimited; and the Holy Ghost unlimited. The Father eternal; the Son eternal; and the Holy Ghost eternal. And yet they are not three eternals; but one eternal. As also there are not three uncreated; nor three infinites, but one uncreated; and one infinite.

So likewise the Father is Almighty; the Son Almighty; and the Holy Ghost Almighty. And yet they are not three Almighties; but one Almighty. So the Father is God; the Son is God; and the Holy Ghost is God. And yet they are not three Gods; but one God.

So likewise the Father is Lord; the Son Lord; and the Holy Ghost Lord. And yet not three Lords; but one Lord. For like as we are compelled by the Christian verity; to acknowledge every Person by himself to be God and Lord; So are we forbidden by the Catholic religion; to say, There are three Gods, or three Lords.

The Father is made of none; neither created, nor begotten. The Son is of the Father alone; not made, nor created; but begotten. The Holy Ghost is of the Father and of the Son; neither made, nor created, nor begotten; but proceeding.

So there is one Father, not three Fathers; one Son, not three Sons; one Holy Ghost, not three Holy Ghosts. And in this Trinity none is before, or after another; none is greater, or less than another. But the whole three Persons are coeternal, and coequal.

So that in all things, as aforesaid; the Unity in Trinity, and the Trinity in Unity, is to be worshipped. He therefore that will be saved, let him thus think of the Trinity.

Furthermore it is necessary to everlasting salvation; that he also believe faithfully the Incarnation of our Lord Jesus Christ.

For the right Faith is, that we believe and confess; that our Lord Jesus Christ, the Son of God, is God and Man; God, of the Essence of the Father; begotten before the worlds; and Man, of the Essence of his Mother, born in the world. Perfect God; and perfect Man, of a reasonable soul and human flesh subsisting. Equal to the Father, as touching his Godhead; and inferior to the Father as touching his Manhood.

Who although he is God and Man; yet he is not two, but one Christ. One; not by conversion of the Godhead into flesh; but by assumption of the Manhood by God. One altogether; not by confusion of Essence; but by unity of Person. For as the reasonable soul and flesh is one man; so God and Man is one Christ;

Who suffered for our salvation; descended into hell; rose again the third day from the dead. He ascended into heaven, he sitteth on the right hand of the God the Father Almighty, from whence he will come to judge the living and the dead. At whose coming all men will rise again with their bodies; And shall give account for their own works. And they that have done good shall go into life everlasting; and they that have done evil, into everlasting fire.

This is the Catholic faith; which except a man believe truly and firmly, he cannot be saved.