What is sedevacantism? It comes from the Latin "sede vacante", which means "vacant chair". Specifically, the vacant chair of Peter the apostle, the first pope. Thus, sedevacantists believe that the "Chair of Peter" (the papacy) is vacant at the current time. So you may well ask, "Why would anyone think that way, when most people in the world recognize John Paul II as pope of the Catholic Church?"
The reason is really fairly simple. The post-Conciliar (and by "post-Conciliar" I mean post-Vatican Council II) "popes" have by their official teaching and bad example in relation to the faith (e.g., John Paul II's allowing himself to be anointed with the "Sign of the Tilak" by a Hindu priestess, attending a "Mass" where the epistle was read by a bare-breasted female and by allowing pagan idols to be set up on Catholic altars at Assisi) have so effectively undermined the Catholic Faith that it can reasonably be said that there is a state of apostasy in the Church today. Note that I am not talking about personal failings deriving from normal human weaknesses here; but only crimes against the faith, which on the part of a pope are simply intolerable.
While there are reasons for doubting bad will on the part of John XXIII and John Paul I (the latter being on the scene so briefly, it is impossible to draw any conclusions about his ultimate intentions), the pontificates of Paul VI and John Paul II are manifest disasters. These two popes, deceptively styled "conservatives" by the liberals, were really the two most radical individuals ever given the helm of the barque of Peter. Both, in their official teaching, have denied doctrine taught by their predecessors (in the Catholic Church truth cannot change); and both, by their personal actions, have given incredible scandal to the faithful. Consequently, there is considerable evidence that these men were (in the case of Paul VI) and are (in the case of John Paul II) seeming heretics. While their teachings make it clear that these men are material heretics, it is more difficult to prove that they fully recognize that their opinions are, in fact, heretical, and thus may be considered formal heretics. Thus, there can be some slight doubt (not much) that they have either never held their offices, or have lost them subsequently. The errors in their official techings clearly indicate that they cannot be under the protection of or guided by the Holy Ghost.
Briefly (as other works on this site will demonstrate clearly) they espouse a theory of religious liberty that is incompatible with the true Faith; and they have modified and corrupted the sacraments of the Church to conform to their false method of universal tolerance in matters of religion to such an extent that it is doubtful that some sacraments continue to be valid. Generally, the object of love in the Church has been turned away from God and toward our fellow men - a new kind of idolatry - cloaked under the alleged pursuit of "love", "universal tolerance" and "inclusion" (buzzwords of the liberals of our day). These and other travesties perpetrated on the part of the post-Conciliar "Catholic" Church (often called the "Conciliar Church" in honor of Vatican Council II by its adherents, as well as many traditional Catholics) will be made clear to you if you read the material on this site.
You may be thinking that "the Holy Spirit promised to be with the Church always, so it is impossible that the pope could lose the faith". It is true that the Holy Spirit will always be with the Church, and on account of that it will endure until the end of the world. But it is not impossible that a pope could become a heretic. St. Robert Bellarmine - a doctor of the Church - in his work "De Romano Pontifice" expresses the opinion that a pope could lose the faith (i.e., become a heretic or apostate) and by that fact would lose his office (his pontifical authority). Other theologians of the late middle ages held similar opinions. It seems that what they feared in their day has clearly come to pass in ours. That this idea is not as outrageous as it may first seem can be shown if you read the text of the file "journet.zip" in the "Miscellaneous Religious Articles" section of this Web site.
However, this opinion of the current state of the papacy falls into the domain of what is called "speculative theology", which includes ideas that the Church has never resolved definitively. Thus, ideas underlying sedevacantism can only truthfully be described as "reasonable" or "probable", but not "certain". The value in the thesis is that it provides moral justification for resistance to the will of those who present themselves to us as popes. Normally, Catholics are obliged to give their full assent to the pope in matters of faith and morals, or be guilty of serious sin, if not heresy. But if the pope himself engages in acts that constitute manifest heresy, it is permissible and reasonable (as indicated above) to question his legitimacy, and thus his authority.
Some people ask "How can you presume to judge other people - especially the pope?" It is important to understand that life requires us to make judgments about the moral impact of many people, circumstances and events every day. Such judgments must be made to ensure and facilitate living a moral life in keeping with the immutable precepts of the faith. Such judgments are PERSONAL, and while they have no value in terms of deposing a bad pope (who can only be self-deposed or deposed by God), they have value in helping those over whom we have responsibility, such as (in the case of parents) our children. Besides, the individual does not really "judge" the pope. This kind of judgment is not simply a matter of expressing a personal opinion. It is a matter of applying Catholic teachings and principles to the current situation in the Church.
Discussion of the problem of an heretical or scandalous pope also has value in dissuading others to join him in his errors. When enough people come to perceive the problem, there is hope that a public rebuke to the apparent pope might take place which might be efficacious in bringing him to his senses. True charity does not consist of encouraging people in their errors but apprising them of the truth.
SEDE List is a place where those of the sedevacantist position (and some others who disagree with the thesis but are interested in it) discuss such issues (and many others besides). If you have been scandalized by some new novelty introduced in your parish, I can only urge you to read the material on this site, then consider joining SEDE List.