Is it her people, her leadership, her beliefs? The term catholic usually means universal, so one would think it's her 1 billion plus people spread throughout the world.
However, I recently pointed out that an overwhelming majority of Catholics use birth control. (link) Does this mean the Catholic Church is OK with birth control? Matthew Bellisario responded by pointing me to an earlier post he had written where he claimed all Christians up till the 1930's rejected birth control.
All Christians up until the 1930s interpreted this text as referring to Onan's punishment of death [Genesis 38 7:9] by his act of “coitus interruptus.” (link)
I responded by quoting Jovinianus' alternative explanation in the 4th century (link).
Matthew then made an interesting move; backing away from his claim of 'all Christians' to 'every Christian group'.
every Christian group before the 1930 interpreted this passage the way I am interpreting it. (link)
So what is a church group? Given that 1) the Catholic church group rejects birth control and 2) most Catholics use birth control, one might think a church group is not really about her people, but rather her leadership. Rome's leadership is so small a group compared to the whole Catholic church that they are statistically insignificant - and in this case they don't represent most Catholics.
Recently Matthew Bellisario added another wrinkle, by posted that Roman Catholics and the Eastern Orthodox are united in Christ due to their common belief of the real presence in the mass. (link) So there's a connection between the East and West that transcends leadership differences; since the East does not recognize the Pope as their leader. So if it's not her people or her leadership, what makes the Catholic church 'Catholic'?
I think Matthew's move here undermines a fundamental principle of Roman Catholicism - namely the epistemic priority of the Church over doctrine. Indeed, for a Catholic to know some doctrine is true, they must first know it's approved by the Church. In some cases knowing the Church teaches something is sufficient to know it's true, without further inquiry. This is of course contrary the the scriptural model of 'search the scriptures' or teach then baptize, but it's also at odds with Matthew's finding unity across Church boundaries, which seems to invite people to examine doctrine first, then look for the Church. How could Rome stand up doctrines such as the assumption of Mary based on the evidence rather than her authority?
Of course, the right answer here is that the catholic church is all those who are assembled to Christ by the call of the Gospel. (Hebrews 12:22-24) Other than that, the bible simply speaks of individual, local congregations as churches.