The following was posted on the Catholic Answers forum on the topic of Peter as the foundation of the church. This quote represents fairly well the Catholic understanding of what the foundation of the church is. Out of everything discussed on that particular thread, this, believe it or not, was the most in-depth any Catholic ever got.
Thou art Peter [Kipha, Cephas] and on this rock [Kipha, Cephas] I will build my Church, cannot be understood save of building the Church on this man Peter (Cephas), otherwise the point of the phrase disappears. Jesus was called the cornerstone (1 Peter 2:4-8; Ephesians 2:20), but He could not be indicating Himself here: it would have been rather like a bad joke, if I may venture to say so: Thou art Peter, but it is on quite another Peter that I am going to build! Some try to return indirectly to this superannuated Protestant interpretation by making out the Rock to be Peter’s faith in the Messiahship of the Lord. It was indeed Peter’s faith that introduced the promise, but the promise is given to the person whose faith has just been displayed. If the building is a group, the foundation is their head: Jesus, says St. John Chrysostom, exalts Peter’s declaration, He made him pastor. The position of Peter in the Church is that of the rock on which the building is erected; thanks to this foundation the building will stand firm; thanks to this head the community will be well ruled.