The Holy Trinity is the greatest mystery of the Catholic and Christian faith. Without even the most basic understanding of the Trinity, the Catholic faith is useless. If Jesus is not God along with the other persons also being God, there is no Catholic faith. Of course there is also no way that any person can fully understand how this can be. Our human minds cannot comprehend a being which is one but is also composed of three distinct persons.
In the Bible
This is a mystery which is only revealed in the New Testament by Jesus, but even in the Old Testament, there are hints that God is more than one person despite being one. “Then God said, 'Let us make man in our image, after our likeness; and let them have dominion over the fish of the sea, and over the birds of the air, and over the cattle, and over all the earth, and over every creeping thing that creeps upon the earth.'” - Genesis 1:26. “Come, let us go down, and there confuse their language, that they may not understand one another's speech.” - Genesis 11:7 These passages from Genesis are notable because God refers to Himself in the plural, yet we know that there is only one God. The only way this makes sense is in the mystery of the Trinity, in which there are multiple persons in one God.
In the New Testament, there are more overt references to the Trinity, which is not surprising since Jesus, Himself part of the Trinity, is the main subject of the New Testament. “And when Jesus was baptized, he went up immediately from the water, and behold, the heavens were opened and he saw the Spirit of God descending like a dove, and alighting on him; and lo a voice from heaven, saying, 'This is my beloved Son, with whom I am well pleased.'” - Matthew 3: 16 – 17. “These things I have spoken to you, while I am still with you. But the Counselor, the Holy Spirit, whom the Father will send in my name, he will teach you all things, and bring to your remembrance all that I have said to you.” - John 14: 25 – 26. “Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.” - Matthew 28:19. In the first passage, the entire Trinity makes an appearance in the Bible at the baptism of Jesus. Of course Jesus is present, and the Holy Spirit descends in the form of a dove, while God the Father speaks to those assembled. In the latter two passages, Jesus mentions all three persons of the Trinity. He tells His disciples on each of these occasions that through the Holy Trinity they will receive grace.
Early Church Fathers
The early Church Fathers also wrote extensively on this subject. Since it is such a mystery, they tried to be as clear as possible, while still teaching truth.
We will prove that we worship him reasonably; for we have learned that he is the Son of the true God himself, that he holds a second place, and the Spirit of prophecy a third. For this they accuse us of madness, saying that we attribute to a crucified man a place second to the unchangeable and eternal God, the Creator of all things; but they are ignorant of the mystery which lies therein. — Justin Martyr First Apology 13: 5 – 6
And at the same time the mystery of the oikonomia is safeguarded, for the unity is distributed in a Trinity. Placed in order, the three are the Father, Son, and Spirit. They are three, however, not in condition, but in degree; not in being, but in form; not in power, but in kind; of one being, however, and one condition and one power, because he is one God of whom degrees and forms and kinds are taken into account in the name of the Father, and of the Son, and of the Holy Spirit. — Tertullian Against Praxeas 2
Nevertheless it seems proper to inquire what is the reason why he who is regenerated by God unto salvation has to do both with Father and Son and Holy Spirit, and does not obtain salvation unless with the co-operation of the entire Trinity; and why it is impossible to become partaker of the Father or the Son without the Holy Spirit. - Origen De Principiis Book 1 Chapter 3
Finally, when, after the resurrection, the apostles are sent by the Lord to the heathens, they are bidden to baptize the Gentiles “in the name of the Father, and of the Son, and of the Holy Ghost.” How, then, do some say, that a Gentile baptized without, outside the Church, yea, and in opposition to the Church, so that it be only in the name of Jesus Christ, everywhere, and in whatever manner, can obtain remission of sin, when Christ Himself commands the heathen to be baptized in the full and united Trinity? - Cyprian Epistle LXXII 5.18
Those holy angels come to the knowledge of God not by audible words, but by the presence to their souls of immutable truth, i.e., of the only-begotten Word of God; and they know this Word Himself, and the Father, and their Holy Spirit, and that this Trinity is indivisible, and that the three persons of it are one substance, and that there are not three Gods but one God; and this they so know that it is better understood by them than we are by ourselves. - Augustine City of God 11.29
On the surface, some of these readings seem nonsensical or confusing, but it is because this mystery is so complicated and incomprehensible. These authors are trying to understand themselves and explain how three persons can all be part of one being, and be equal but distinct. The story of St. Augustine and the Trinity comes to mind. St. Augustine was walking on the beach, thinking about the Trinity, and he happened upon a boy scooping water into a hole. He learned that the boy was trying to put the ocean into the hole, and he told the boy that this was impossible. The boy then turned into an angel and told Augustine that it would be easier to put the ocean in a hole than to comprehend the Trinity completely. We accept this truth on faith and try to understand it as best as we can with out limited brains.