The Feast of the Conversion of Saint Paul the Apostle celebrates the conversion told in several of Saint Paul's epistles and in the Acts of the Apostles. The feast comes at the end of the Week of Prayer for Christian Unity and celebrates the conversion as proof that "no fall so deep that grace can not descend to it" (Johann Peter Lange, A commentary on the Holy Scriptures, Volume 8).
The Apostle Paul describes this conversion two epistles - the First Epistle to the Corinthians and the Epistle to the Galatians. In the Epistle to the Galatians, Saint Paul describes himself as a Pharisee of Pharisees, writing (Epistle of Saint Paul to the Galatians 1:13-14, Douay-Rheims Bible):
 For you have heard of my conversation in time past in the Jews' religion: how that, beyond measure, I persecuted the church of God, and wasted it.  And I made progress in the Jews' religion above many of my equals in my own nation, being more abundantly zealous for the traditions of my fathers.
Furthermore, in the Acts of the Apostles, Saint Paul describes his participation in the stoning of Saint Stephen (Acts of the Apostles 7:56-58, Douay-Rheims Bible):
 And they crying out with a loud voice, stopped their ears, and with one accord ran violently upon him.  And casting him forth without the city, they stoned him; and the witnesses laid down their garments at the feet of a young man, whose name was Saul. And they stoned Stephen, invoking, and saying: Lord Jesus, receive my spirit.
The conversion itself takes place on the road to Damascus, as described in the Acts of the Apostles and the two epistles (Acts of the Apostles 9:1-20, Douay-Rheims Bible):
 And Saul, as yet breathing out threatenings and slaughter against the disciples of the Lord, went to the high priest,  And asked of him letters to Damascus, to the synagogues: that if he found any men and women of this way, he might bring them bound to Jerusalem.  And as he went on his journey, it came to pass that he drew nigh to Damascus; and suddenly a light from heaven shined round about him.  And falling on the ground, he heard a voice saying to him: Saul, Saul, why persecutest thou me?  Who said: Who art thou, Lord? And he: I am Jesus whom thou persecutest. It is hard for thee to kick against the goad.
 And he trembling and astonished, said: Lord, what wilt thou have me to do?  And the Lord said to him: Arise, and go into the city, and there it shall be told thee what thou must do. Now the men who went in company with him, stood amazed, hearing indeed a voice, but seeing no man.  And Saul arose from the ground; and when his eyes were opened, he saw nothing. But they leading him by the hands, brought him to Damascus.  And he was there three days, without sight, and he did neither eat nor drink.  Now there was a certain disciple at Damascus, named Ananias. And the Lord said to him in a vision: Ananias. And he said: Behold I am here, Lord.
 And the Lord said to him: Arise, and go into the street that is called Strait, and seek in the house of Judas, one named Saul of Tarsus. For behold he prayeth.  (And he saw a man named Ananias coming in, and putting his hands upon him, that he might receive his sight.)  But Ananias answered: Lord, I have heard by many of this man, how much evil he hath done to thy saints in Jerusalem.  And here he hath authority from the chief priests to bind all that invoke thy name.  And the Lord said to him: Go thy way; for this man is to me a vessel of election, to carry my name before the Gentiles, and kings, and the children of Israel.
 For I will shew him how great things he must suffer for my name's sake.  And Ananias went his way, and entered into the house. And laying his hands upon him, he said: Brother Saul, the Lord Jesus hath sent me, he that appeared to thee in the way as thou camest; that thou mayest receive thy sight, and be filled with the Holy Ghost.  And immediately there fell from his eyes as it were scales, and he received his sight; and rising up, he was baptized.  And when he had taken meat, he was strengthened. And he was with the disciples that were at Damascus, for some days.  And immediately he preached Jesus in the synagogues, that he is the Son of God
And in the First Epistle to the Corinthians (1st Epistle to the Corinthians 15:1-8, Douay-Rheims Bible):
 Now I make known unto you, brethren, the gospel which I preached to you, which also you have received, and wherein you stand;  By which also you are saved, if you hold fast after what manner I preached unto you, unless you have believed in vain.  For I delivered unto you first of all, which I also received: how that Christ died for our sins, according to the scriptures:  And that he was buried, and that he rose again the third day, according to the scriptures:  And that he was seen by Cephas; and after that by the eleven.
 Then he was seen by more than five hundred brethren at once: of whom many remain until this present, and some are fallen asleep.  After that, he was seen by James, then by all the apostles.  And last of all, he was seen also by me, as by one born out of due time.
And in the Epistle to the Galatians (Epistle to the Galatians 1:15-16, Douay-Rheims bible):
 But when it pleased him, who separated me from my mother's womb, and called me by his grace,  To reveal his Son in me, that I might preach him among the Gentiles, immediately I condescended not to flesh and blood
Father Roger J. Landry, in his homily on Saint Paul's Conversion (January 25th, 2014 http://catholicpreaching.com/st-pauls-conversion-and-ours-january-25-2014/) writes:
Paul’s conversion was, rather, from a false notion that we are saved by our external adhesion to all the prescriptions of the Mosaic Law, to the true one that we are saved by grace through faith in Jesus Christ. We are saved by Christ’s work, not our own. The culmination of the saving life of faith he wrote about in his letter to the Galatians when he said, “I have been crucified with Christ and the life I now live I live by faith in the Son of God who loved me and gave himself up for me” (Gal 2:20). The true notion of holiness is to die to ourselves so that the Risen Christ truly can live within us, reign within us, sanctify and save us and make us his instruments to co-redeem the world. Holiness is union with God. Since we are saved by grace, and grace is not a thing but a participation as a creature in the life of the Creator, Christian conversion must be continual, because it’s based on a continued encounter with the Lord, as he seeks in us to form us more and more in his image with our free fiat. In St. Paul’s life we see that conversion was not a one-time thing but a continuous reality as he continued to grow in the Gospel that he was fearlessly and faithfully proclaiming.