There were a few antipopes in the early Church, typically related to a division based on other doctrines. For example, Hippolytus of Rome himself (a fairly well-known figure in the early Church and a valuable historical source for that period) at one point led a rival faction against the bishop now considered by the Roman Catholic Church to be the rightful Pope at the time, but apparently rejected that at some point and was later reconciled to that bishop, because after his death he was recognized as a martyr and was stated to be a priest (but not bishop). Another example of an early antipope was Novatian, the originator of the movement now aptly named Novatianism and considered heretical by the Catholic Church; Novatian thought the current pope was too lenient on Christians who had lapsed and denied the faith during 3rd century Roman persecution, and so broke away and was recognized as a bishop of Rome and rival to the other.

The "golden age" of antipopes was, however, the Middle Ages, and unlike the early Church examples, these were motivated more by politics than by doctrine. The prime example are the popes between 1378 and 1437. For much of the 14th century, the popes had moved from Rome to Avignon in France, but in 1377 Pope Gregory XI decided to return to Rome - a decision not popular with the French clerics. When Gregory died the following year, an Italian conclave elected a Roman successor, but he quickly turned out unsatisfactory to many of the cardinals, so another conclave was soon held by mostly French cardinals, who elected another pope reigning in Avignon. This state of affairs continued for over thirty years, with each rival papal court electing successors as necessary, until a church council met in Pisa in 1409 to resolve the issue.

Unfortunately, their solution was to depose both papal claimants, in Rome and Avignon alike, and to elect a third man. Unsurprisingly, neither of the previous popes recognized this decision, so for a time there were three papal claimants - depending on who you supported, one pope and two antipopes.

The schism was finally (officially) resolved in 1417 at the Council of Constance, a council recognized by both the Pisan and Roman candidates. The council decided that all three popes (Pisa, Rome, and Avignon) should be removed and one final replacement elected. The Roman claimant, Gregory XII, resigned and is credited by the Catholic Church as being the last legitimate pope to resign his office before Benedict XVI in 2013, the predecessor of the current Francis. The Pisan claimant fled, and the Avignon claimant (who, unlike the other two, had not recognized Constance as a legitimate council) persisted for a while, with successors until 1437, but with rapidly diminishing influence and for all practical purposes the schism was settled in 1417.

The 15th century is the end of the "historical" antipopes, but there are still certain claimants today. There are a variety of small splinter groups sometimes called "sedevacantist", a word meaning "the seat is vacant" - they typically see the decisions of the Second Vatican Council in the 1960s as illegitimate and denying the historic Roman Catholic faith, and therefore the popes since then are likewise illegitimate; hence the name, "the seat (of Peter as pope) is vacant". For example, there's an American who claims the title Pope Michael. I believe he has perhaps a few dozen followers; since the 15th century, there haven't been any serious antipope candidates who were recognized by large swaths of the Catholic world.