Conclavism is the claim to election as pope by a group acting or purporting to act in the stead of (i.e., under an assumption of the authority ordinarily vested in) the established College of Cardinals. This claim is usually associated with the claim, known as sedevacantism, that the present holder of the title of pope is a heretic and therefore not truly pope, as a result of which the faithful remnant of the Catholic Church has the right to elect a true pope.
The term comes from the word "conclave", the term for a meeting of the College of Cardinals convened to elect a Bishop of Rome, when that see is vacant, but which proponents of conclavism apply to the group that elects an alternative pope.
A similar but distinct phenomenon is that of those (referred to as "mysticalists") who base their claim to the papacy on supposed personal supernatural revelations.
The phenomenon of sedevacantism developed in the late 1960s and the 1970s, the years that followed the Second Vatican Council. In the mid-1970s, the sedevacantist pioneer Father Joaquín Sáenz y Arriaga of Mexico advocated holding a papal election, and some other traditionalist Catholics discussed the idea in the following years. However, conclavism became an actual movement only in the 1990s.
The first to claim (in 1978) to have been elected Pope in this way was the Croatian, Mirko Fabris, a stand-up comic who performed under the jocose name "Krav" (a masculinization of the feminine noun krava, meaning "cow") and who accordingly called himself Pope Krav I.
Meant more seriously was the claim of David Bawden, who in the late 1980s promoted the idea of a papal election and ultimately sent out over 200 copies of a book of his to the editors of all the sedevacantist publications he could find, and to all the priests listed in a directory of traditionalists as being sedevacantists. He was then elected by a group of six people who included himself and his parents, and took the name "Pope Michael".