The name John has been used by 23 popes.
There is no rule that prohibits a pope from taking the name of Peter. However, out of respect for the first pope, no man has used that name.
Benedict is the name used by 15 popes. Pope Benedict X was an antipope and not counted as a legitimate pope.
There are a number of pope names that have been used only once. Peter, the name of the first pope, is one and Francis, the name of the current (2013) pope, is another example.
Innocent has been the name of 13 popes I believe.
Papal bull
The most common name used by popes is John. There have been 23 popes named John. That is followed by Gregory with 16. Benedict has been used by 15 popes. There was no legitimate Pope Benedict X. He as an anti pope.
The skull cap worn by a pope, cardinal or bishop is called a zucchetto.
A pope does not have to change his name but most do. It is symbolic of Jesus changing the name of Simon to Peter at the time he named him as leader of his Church and the first pope.
"They" did not change the name of the new pope, When elected, the new pope has the choice of keeping his Christian name or choosing a regnal name that he wishes to be known as while serving. When Christ named Simon as the first pope he changed his name to Peter. After Peter, the next pope to choose a new name was Pope John II who was elected in the year 533. His given name was Mercurius and he did not think it proper for a Catholic pope to be named for a pagan Roman god - Mercury.
I could name 266 popes but will name the most 2 recent ones: Pope Benedict XVI and Pope Francis.
There were about 20 popes with the name Gregory, be more specific eg Pope Gregory iV
The words pope, archbishop and bishop are capitalized when they are used as part of the man's name. Such as Pope Pius X, or Bishop O'Connor.