What is a Byzantine Catholic Church?

The Catholic Church is a communion of churches. It is made up of churches from the Eastern Tradition and the Western Tradition. Eastern Catholics are in union with Rome. We share the same basic faith and the same mysteries (sacraments), however, our way of expressing them follows the same tradition as the Orthodox churches. In reality, there are many Eastern churches, each with its own heritage and theology, liturgy and discipline.
Jesus sent his disciples to the four corners of the world to spread the Gospel. Eventually, four great centers of Christianity emerged with distinctive Christian customs, but the same faith. These centers were Jerusalem, Antioch, Rome and Alexandria. A few centuries later when the capital of the Roman Empire was moved to the Eastern city of Byzantium, later renamed Constantinople, an adaptation of the Antioch celebration of the liturgy was made.
From this powerful cultural center the Byzantine church emerged.
(Joseph Radvansky, A Brief Explanation of the Eastern Catholic Churches, Introduction)
About the Ukrainian Catholic Church

Eastern Christianity took a firm root in Ukraine in 989 when Vladimir, Prince of Kiev, embraced the Christian Faith and was baptized. Soon afterwards many missionaries from the Byzantine Empire arrived, having been sent by the Patriarch of Constantinople to preach the Gospel.
When the Church of Rome and the Church of Constantinople severed ties with one another in the 11th century, the Church in Ukraine gradually followed suit and finally gave up the bonds of unity with Rome. When Ukrainian Orthodox bishops met at a council in Brest-Litovsk in 1595, seven bishops decided to re-establish communion with Rome. Guaranteed that their Byzantine tradition and Liturgy would be respected and recognized by Rome, they and many priests and lay faithful were re-united with the See of Rome, while others continued to remain Orthodox.
In the 19th century many Ukrainian Catholics began to emigrate to North America, bringing their pastors, traditions and liturgy to Canada and the United States. Under Communist rule, Catholics in Ukraine were persecuted, with many being imprisoned and murdered; in 1945 all the Ukrainian Catholic bishops were arrested or killed. (To read about the New Martyrs of Ukraine, click here.)
Today the Ukrainian Catholic Church is the largest Eastern Catholic Church, with about 5 million faithful. It is led by His Beatitude Sviatoslav Shevchuk, Major Archbishop of Kyiv-Galicia. His election was confirmed by Pope Benedict XVI on March 25, 2011.
(excerpted and adapted from nativityukr.org)
About St. Thomas the Apostle

Thomas was one of the Twelve Apostles. Through his doubt in the Resurrection of Christ the Lord, a new proof was given of that wonderful and saving event. The resurrected Lord appeared to His disciples a second time, in order to convince Thomas. The Lord said to Thomas: Reach hither thy finger, and behold My hands; and reach hither thy hand, and thrust it into My side: and be not faithless, but believing. And Thomas replied: My Lord and my God (John 20:27-28). After the descent of the Holy Spirit, when the apostles cast lots to see where they would each go to preach, the lot fell to Thomas to go to India. He was a little saddened that he had to go so far away, but the Lord appeared to him and encouraged him. In India, St. Thomas converted many, both aristocrats and poor, to the Christian Faith, and established the Church there, appointing priests and bishops. Among others, Thomas converted two sisters to the Faith–Tertiana and Migdonia–both wives of Indian princes. Because of their faith, both sisters were ill-treated by their husbands, with whom they no longer wanted to live after their baptism. Eventually, they were allowed to go. Being freed of marriage, they lived God-pleasing lives until their repose.
Dionysius and Pelagia were betrothed, but when they heard the apostolic preaching they did not marry, but devoted themselves to the ascetic life. Pelagia ended her life as a martyr for the Faith, and Dionysius was ordained a bishop by the apostle. Prince Mazdai, Tertiana’s husband, whose son, Azan, was also baptized by Thomas, condemned the apostle to death. Mazdai sent five soldiers to kill Thomas. They ran him through with their five spears, and thus the Holy Apostle Thomas rendered his soul into the hands of Christ. Before his death, he and the other apostles were miraculously brought to Jerusalem for the burial of the Most-holy Theotokos. Arriving too late, he wept bitterly, and the tomb of the Holy Most-pure One was opened at his request. The Theotokos’ body was not found in the tomb: the Lord had taken His Mother to His heavenly habitation. Thus, in his tardiness St. Thomas revealed to us the wondrous glorification of the Mother of God, just as he had once confirmed faith in the Resurrection of the Lord by his unbelief.
(St. Nikolaj Velimirović, The Prologue of Ochrid, October 6th)