The term antipope refers to any person who claims to be pope, but whose claim is treated as invalid today by the Roman Catholic Church. This should be a straightforward concept, but in practice it is much more difficult and complex than it might appear.
The problems lie in determining who qualifies as pope and why. It isn't enough to say that their election did not follow standard procedures, because those procedures have changed over time. Sometimes the rules arent even followed. Innocent II was elected in secret by a minority of cardinals but his papacy is treated as legitimate. It isn't enough to say that an alleged pope did not lead an adequately moral life because many legitimate popes led terrible lives while the first antipope, Hippolytus, is a saint.
Over time names have shifted back and forth between the lists of popes and antipopes because people have changed their minds about what to do with them. The Vaticans official list of popes is called the Annuario Pontificio and even today there remain four instances where it isnt absolutely clear on whether someone was a legitimate successor of Peter.
Silverius vs. Vigilius: Pope Silverius was forced to resign by Vigilius who became his successor, but the dates dont match up properly. The date of Vigilius election is listed as March 29, 537, but Silverius resignation is marked as November 11, 537. Technically there cant be two popes at the same time, so one of them had to be an antipope but the Annuario Pontificio treats them both as valid popes for the time period in question.
Martin I vs. Eugenius I: Martin I died in exile on September 16, 655, without ever having resigned. The people of Rome werent sure that he would return and didnt want the Byzantine emperor to impose someone awful on them, so they elected Eugenius I on August 10, 654. Who was the real pope during that year? Martin I was not removed from office by any canonically valid procedure, so Eugenius election should be treated as invalid but hes still listed as a legitimate pope.
John XII vs. Leo VIII vs. Benedict V: In this very confusing state of affairs, Leo was elected pope on December 4, 963, while his predecessor was still alive John didnt die until May 14, 964 and he never resigned. Leo, in turn, was still alive when his successor was elected. Benedicts papacy is listed has having started on May 22, 964 (just after the death of John) but Leo didnt die until March 1, 965. So, was Leo a legitimate pope, even though John was still alive? If not, then Benedict was presumably valid, but if he was, then how was Benedict a valid pope? Either Leo or Benedict has to have been an invalid pope (an antipope), but the* Annuario Pontificio doesnt decide one way or the other.
Benedict IX vs. Everyone Else: Benedict IX had the most confusing papacy, or the most confusing three papacies, in the history of the Catholic Church. Benedict was forcibly removed from office in 1044 and Sylvester II was elected to take his place. In 1045 Benedict seized control again, and again he was removed but this time he resigned as well. He was succeeded first by Gregory VI and then by Clement II, after which he returned once again for a few months before being ejected. Its not clear that any of the times Benedict was removed from office was canonically valid, which would mean that the other three mentioned here were antipopes, but the Annuario Pontificio continues to list them as genuine popes.