Also sometimes known as Jude Lebbeus or just Thaddeus, Jude Thaddeus was the brother of another apostle, St. James the Less and he is the "mystery" apostle because he's the one the synoptic gospels disagree on. Mark and some versions of Matthew list him as Thaddeus; some versions of Matthew list him as Lebbeus; some versions of Matthew list him as Judas the Zealot; Luke lists him as Judas, son of James.
It is assumed that these are all the same person and it's reasonable to think that some confusion resulted from efforts to distinguish him from Judas Iscariot. Jude Thaddeus certain doesn't appear very often in the New Testament stories. He is depicted as asking Jesus why he doesn't appear before the whole world after his resurrection, but otherwise he only shows up when the names of various apostles are listed. The Epistle of Jude was originally attributed to him, but modern scholars generally date this work to the first quarter of the 2nd century and was not likely written by someone who was an adult during the first quarter of the 1st century.
With so little scriptural information about Jude Thaddeus, legends quickly developed. There was much demand from early Christians details about what happens to Jesus' companions and Christian leaders complied. Some said that he was the bridegroom for the wedding at Cana. Others said that he traveled to places like Syria, Libya, and Mesopotamia to preach the gospel.
Some traditions say that he was martyred in Persia and that his body was placed in a crypt in St. Peter's Basilica. Other traditions say that he brought Christianity to Armenia with Saint Bartholomew and that he was martyred in Armenia. There is no independent evidence for any of this.
Thaddeus is also the name of one of the "Seventy Apostles of Christ," a group of followers mentioned in Luke 10:1-20 who were appointed by Jesus to go out ahead of him and preach the gospel. This Thaddeus is believed to have been a Jew born in Edessa. According to Eusebius, he was martyred in Mesopotamia in 44.