Well, first of all, rank is more associated with the people in the hierarchy and not in the church buildings themselves.
With the people in the hierarchy, the world is divided into dioceses (a ‘diocese’ was a province in the Roman Empire). Each diocese is headed by a bishop. The bishop of Rome is the Pope and the head bishop.
A diocese is divided into parishes. The head of the parish is a priest designated as pastor (there are lots of other priests who are not pastors – they are assistants, teachers, missionaries, etc…).
So, it goes Pope (Bishop of Rome) --> Bishop —> Pastor (who’s a priest).
Now, priests and bishops represent two of the three Holy Orders. The third is deacon, and heirarchically, deacons have no real power in the line of authority unless they’ve been designated with it on a case by case basis.
Now, some dioceses are designated as archdioceses. Usually, they were the parent diocese from which the other surrounding dioceses were split off from. The archdiocese in a local region of diocese has no real authority over the regular dioceses. Although, the bishop of an archdiocese is called an archbishop and it is a prestigious assignment for a bishop and puts him on the short list to be made a cardinal.
A cardinal is someone designated to be the elector of the next Pope. A cardinal is almost always an archbishop, and the title confers no real authoritative power over other bishops.
Now, getting back to the OP:
In each diocese the bishop (or archbishop or Pope) has his ‘home parish’ and the church building is called a cathedral. This comes from the latin word cathedra which means seat. The church has a special seat which only the bishop uses which represents his ‘seat of power.’
So, the bishop’s church, the cathedral, outranks the other churches in the dioceses, even if it’s a simple building. The archbishop’s cathedral outranks the other cathedrals in the province. Older cathedrals outrank younger ones. The Pope’s cathedral wins.
All the other parish churches within a diocese have equal ranking (although fussbudgets will argue that older parishes outrank younger ones).
Now, there’s a wrinkle in all this. The basilica. Basically, a basilica is: 1) a style of architecture; and, 2) an honorary title. As a title, it bumps a church up ahead of the other churches in the diocese but still behind the cathedral.
Check out: http://www.newadvent.org/cathen/02325a.htm