The Catholic Church in Rome was founded upon the labor of Peter. Paul also did work there, but he was careful not to tread upon the foundation Peter had already laid.

"And I have so preached this gospel, not where Christ was named, lest I should build upon another man's foundation."
- Romans 15:20

We can also see that Paul only began his ministry after meeting with Peter despite already having a vision of Jesus.

"Then, after three years, I went to Jerusalem, to see Peter, and I tarried with him fifteen days."
- Galatians 1:18

We can see from Peter's epistle in the New Testament that Peter led the Catholic Church in Rome.

"The church that is in Babylon [Rome], elected together with you, saluteth you: and so doth my son Mark."  
- 1 Peter 5:13

But did Peter lead the whole Church on earth as Christ seemed to indicate? For example, when Jesus told Peter to "feed my sheep" (Jn 21:15-17) or when he built the Church on Peter (Mt 16:18), or when he had Peter pay Jesus' tax (Mt 17:23).

Let's see what happened in the Acts of the Apostles:

By word of Peter a new bishop/apostle was chosen. (Acts 1:20-26)

By word of Peter judgment is passed on Ananias and his wife for their incorrect behavior as disciples. (Acts 5:1-6)

By word of Peter a doctrinal issue is resolved at the Apostolic Council. After Peter spoke, all were silent. Only afterward do Paul and Barnabas speak in support. (Acts 15:7-12)

Some would point to the Apostolic Council as showing Peter did not exercise authority, but this is misguided. 

The proceedings at the Apostolic Council were modeled on the only experience they had for resolving doctrinal issues, which was procedurally the way the Sanhedrin worked. The person presenting the tradition was the decisive actor for the question at hand. 

"When a question was raised and a member of the college declared that he was in possession of a tradition according to which the question might be decided, such tradition was decisive."
- Tosef., Sanh. vii. 1

Peter was the decisive actor at the Apostolic Council by presenting the tradition. After Peter spoke, all were silent. (Acts 15:12)

James was Catholic bishop of Jerusalem, and it was his duty to preside at the council. He, like Paul, and Barnabas only joined their judgement with Peter and the decisive tradition that Peter was presented. 

It should also be noted that Peter was not trying to Judaize the Christian faith by incorporating the Jewish law and customs. He was merely trying not to scandalize the potential Jewish converts that he was evangelizing when he observed Jewish customs in their presence. Peter's influence was so great that even Paul's disciple Barnabas emulated Peter before the Apostolic Council (Gal 2:13). After the council, we can see that Paul started to emulate Peter, as we hear Paul state:

"And I became to the Jews, a Jew, that I might gain the Jew"
- 1 Cor 9:20