A regnal name, or reign name, is a formal name used by some popes and monarchs during their reigns. Since medieval times, monarchs have frequently chosen to use a name different from their own personal name (and therefore the regnal name is technically a pseudonym) when they inherit a throne.
The new name (or sometimes the old one, confirmed) is followed by an ordinal to give a unique name for the period when the monarch is on the throne. However, in the case of a personal union, the same ruler may carry different ordinals in each state, as they are each assigned chronologically; but some may have more precursors of the same Christian name (usually from a different dynasty).
In parts of Asia, monarchs take era names. Even where that is not the case, rulers may — instead of a whole dynasty, as is the case with Georgian, referring to several Georges of the Hanoverian dynasty — become eponymous of their age, e.g. in Britain: Victorian (even applied to the rest of the world, and less correctly to its alleged prudish mentality) or Edwardian.

Ancient rulers
In parts of Asia, it is more a rule than an exception that monarchs take additional names when ascending, and quite often discard the name they were known by as princes. Often the assumed name is different from his childhood name, and a new temple name could be assumed. A posthumous name is sometimes accorded to a deceased monarch. See, for example, the list of Emperors of Japan.

East Asia
Immediately after a new pope is elected, and accepts the election, he is asked by the Dean of the College of Cardinals, "By what name shall you be called?" The Pope-elect chooses the name by which he will be known from that point on. The senior Cardinal Deacon, or Cardinal Protodeacon, then appears on the balcony of Saint Peter's to proclaim the new Pope, informing the world of the man elected Pope, and under which name he would be known during his reign.
Annuntio vobis gaudium magnum: Habemus Papam! Eminentissimum ac Reverendissimum Dominum, Dominum [forename], Sanctæ Romanæ Ecclesiæ Cardinalem [surname], qui sibi nomen imposuit [papal name].
("I announce to you a great joy: We have a Pope! The Most Eminent and Most Reverend Lord, Lord [forename], Cardinal of the Holy Roman Church [surname], who takes to himself the name [papal name].")
Although the apostle Simon was given a new name, Peter, when Jesus declared that he would be the rock (original greek in which the Gospels were written-petros) on which He would build His Church, the Pope choosing a regnal name as a matter of course would not come until later. During the first centuries of the church, men elected Bishop of Rome continued to use their baptismal names after their elections. The custom of choosing a new name began in AD 533 with the election of Mercurius. Mercurius had been named after the Roman god Mercury, and decided that it would not be appropriate for a pope to be named after a Roman god. Mercurius subsequently decreed that he would be known as John II. Since the end of the tenth century the pope has customarily chosen a new name for himself during his Pontificate; however, until the 16th century some men used their baptismal names.
The last pope to use his baptismal name was Pope Marcellus II in 1555. The names chosen by popes are not based on any system other than general honorifics. They have been based on immediate predecessors, mentors, political similarity, or even after family members—as was the case with Pope John XXIII. The practice of a man using his baptismal name as pope has not been ruled out and future popes could elect to continue using their baptismal names after being elected pope.
Often the man's choice of name upon being elected to the papacy is seen as a signal to the world of who the new pope will emulate, what policies he will seek to enact, or even the length of his reign. Such is the case with Benedict XVI - it was speculated that he chose the name because he wished to emulate the last Pope Benedict, and to also call attention to the fact that at 7.5 years that Benedict XV's reign was a relatively short one.
There has never been a Pope Peter II. Even though there is no specific prohibition against doing so, men elected to the Papacy have refrained from doing so. This is because of a tradition that only Saint Peter should have that honor. In the 10th century John XIV used the regnal name John because his given name was Peter. While some antipopes did take the name Peter II, their claims are not recognized by the mainstream Roman Catholic Church, and each of these men only either has or had a miniscule following which recognized their claims.
Probably because of the controversial Antipope John XXIII, men avoided taking the regnal name John for over 600 years until the election of John XXIII. Immediately after John's election as Pope in 1959, there was some confusion as to whether he would be known as John XXIII or John XXIV. John put the issue to rest by stating that he would be known as John XXIII.
In 1978, Albino Luciani became the first pope to use two names for his regnal name when he took the name John Paul I. He did this to honor both John XXIII and Paul VI. With the unexpected death of John Paul I a little over a month later, Karol Wojtyła took the name John Paul II to honor his immediate predecessors.
Antipopes also have regnal names, and also use the ordinal to show their position in the line of previous pontiffs with their names. For example, Lucian Pulvermacher took the name Pius XIII when he had been named pope by the True Catholic Church sect. If a future legitimate pope takes the name that the antipope used, the antipope would have no bearing on what ordinal the new pope used. Such would be the case if a future pope took the name Pius, as he would then be known as Pius XIII.