Sedevacantism

Sedevacantism is the belief that the papal office is currently vacant, that any person claiming to fill such office was not validly elected, and that the Roman Catholic Church is currently without a sitting Pope. More specifically, it refers to the traditionalist Catholic belief that all popes since the 1958 death of Pius XII are illegitimate. That means John XXIII (or Paul VI) and succeeding popes were and are pope in name only. Despite the non-acceptance of the Pope being the most visible factor, the issues are generally broader and typically center around reforms in the Roman Catholic Church instituted during and after the Second Vatican Council.

Conspiracy theories
While sedevacantism does not per se encompass conspiracy theories, sedevacantists seem prone to following them. Freemason and Communist infiltrators are often seen as the reason these Popes got elected. Conspiracies concerning Communism and Jewish Kabbalah are not uncommon either. The modern Roman Catholic Church is seen as "run by anti-Catholic infiltrators" doing everything they can to wreck real Catholic faith.

Use as snarl word
Some Christian sects, such as the Society of St Pius X, often accuse other sects of being sedevacantists (in longwinded articles). This is ironic since they themselves typically just follow the ideas of the pope they personally like, thus effectively making no difference to a degree. However, the SSPX does begrudgingly accept certain "modernizations" such as the current rules of fasting and abstinence. They simply recommend the older practices.

Ideology
The objections to Francis and his immediate predecessors range from the silly (recent popes have embraced the heresy of Modernism, and no heretic can be a true pope) to the very silly (you can't be a real pope if you don't wear the papal tiara). These objections mainly concern theological and liturgical issues. Immoral behaviour of historical popes, such as hosting or, is not considered to make their papacy invalid.

Sedevacantists typically believe that the Second Vatican Council was illegitimate due to its inclusion of ecumenism and because it replaced the traditional Mass established at the Council of Trent (said in Ecclesiastical Latin and seen as "an expression of the pure Catholic Faith"). The Second Vatican Council illegitimately introduced instead the Novus Ordo Mass (typically said in modern languages and seen as an expression "of the poisonous new modernistic faith" of John XXIII and later popes specially Paul VI).

Beliefs
Typically, sedevacantists believe the following:
 * The last clearly orthodox Pope was Pius XII. Some sedevacantists claim that John XXIII, who convened Vatican II, was a modernist heretic and therefore not really Pope, as no heretic could become a true Pope. Others accept John XXIII as a Pope, but accuse Paul VI of heresy.  Alternatively, he showed his true nature and fulfilled some bullshit "prophecy" when he laid his papal tiara on the altar of St. Peter's Basilica.
 * Since the Second Vatican Council, the accepted hierarchy of the Catholic Church (which they refer to as the Conciliar Church) has promoted "indifferentism" (the idea that there are multiple, mutually acceptable paths to salvation). Sedevacantists adhere to the doctrine of extra ecclesiam nulla salus:  outside the Catholic Church there can be no salvation.  In practice, this means they tend to include any calls by the Catholic hierarchy for religious toleration as "indifferentism."
 * After the Council of Trent, the Tridentine Mass became the only acceptable form of the Mass in the for all time and could never be changed.  Thus, Vatican II's changes to the Mass simply proved that it was in error, illegitimate, and should be ignored.
 * Various nutty conspiracy theories about the Third Secret of Fatima, sometimes involving the "murder" of Pope John Paul I.

Sedeprivationism
Sedeprivationists believe that the Popes since Vatican II have not been full Popes, only a sort of 'potential Pope' or 'half-Pope'. They believe that Francis is a heretic, and as such could not really be Pope; but that if he had recanted his heresy, he would automatically have become Pope the moment he did so.

Conclavism
Closely related to (and funnier than) sedevacantism is conclavism. Basically, conclavists are sedevacantists who have elected their own popes. There are several competitors for the claim to be the "real pope," but one of the most noted conclavist "antipopes" is David Bawden of Belvue, Kansas (aka Pope Michael). A particularly crazy form of conclavism is mysticalism, whose adherents believe their claim to the papacy is received from personal divine revelations or apparitions without a need for a conclave election.

Siri thesis
Belief that the conservative Cardinal Giuseppe Siri was elected Pope in the 1958 papal conclave, but that his election was then suppressed by Zionists and Freemasons, making all subsequent papal claimants antipopes and impostors. Siri himself did not adhere to this claim.

"Benedict is my pope"
Various traditionalists prefer the abdicated Benedict XVI to his replacement Pope Francis. Far-rightists such as the Northern League oppose Francis's support for refugees and immigrants, and wear t-shirts with the slogan "Benedict is my pope". Benedict hasn't helped Francis's authority by interfering and criticising his successor.