What specific authority did he have?
He did not run a church.
He was like the other Apostles a missionary, an evangelist.
The authority that St. Paul talks about when he writes, "even though as apostles of Christ we might have asserted our authority." (1 Thess 2:6)
"For even if I boast somewhat further about our authority, which the Lord gave for building you up and not for destroying you, I will not be put to shame," (2 Cor 10:8)
"For this reason I am writing these things while absent, so that when present I need not use severity, in accordance with the authority which the Lord gave me for building up and not for tearing down" (2 Cor 13:10)
It is also the authority that Simon Magus wanted, when he said to St. Peter and St. John: "Give this authority to me as well, so that everyone on whom I lay my hands may receive the Holy Spirit." (Acts 8:19)
And here St. Paul shows that the Apostles received their authority from Christ: "For you know what commandments we gave you by the authority of the Lord Jesus." (1 Thess 4:2)
Titus, who was a bishop in Crete, received this authority from St. Paul. This is why St. Paul wrote to him: "These things speak and exhort and reprove with all authority. Let no one disregard you." (Titus 2:15) That's an example of Apostolic succession.
This authority included the authority to forgive and retain sins (John 20:23), the authority to confirm (Acts 8:14-19), the authority to ordain (Acts 6:6), the authority to determine doctrine and practice for all the Church (Acts 15), and the authority to bind and loose (Matt 16:19; 18:18). The authority to bind and loose means that whomever the Apostles (and their successors) exclude from their communion, will be excluded from communion with God; whomever they receive anew into their communion, God will welcome back into His.
-A8