The Catholic Religion is unique in that it remains fundamentally intact despite disagreements in doctrine, authority and policy that have arisen over the centuries. Usually, when such disagreements arise they are addressed at a local level, and then perhaps a regional level. If they persist and spread eventually a bishop may issue an edict that reaffirms Catholic belief in the matter. If things are too complex or foregone for even this, the Church will call a Council to examine what is transpiring and to make a definitive decision. After this decision any intransigent parties are not tolerated, lest they corrupt the Faith with their novel teachings or bad will. At this point, the dissents break off and either start their own religion/church or they cease practicing anything at all and may even resort to persecuting the Church.
From the perspective of the Catholic Church, the Orthodox are Catholics who are in schism (and vice versa from the Orthodox perspective). This means that the Orthodox do not recognize the pope as the head of the Church and as such, they are Catholics who are outside of the Church who have usurped Church mandate without the proper authority, that is, the pope to legitimize their actions and belief. If the Orthodox accepted the pope as the head they would become part of the Church once more, since almost all of their doctrine is intact since the split and their sacraments and clergy are validly ordained and administered.
Even more broadly, Catholics consider all the baptized to be Catholic since the Church believes there is only one baptism, Christ's, and as Christ's Church on Earth the sacrament belongs to Christ's Church. Thus anyone who is validly baptized the Church sees as baptized Catholic. To this end, all Protestants are baptized Catholics who profess heretical beliefs or are in schism or both, though it must be admitted, after the many centuries and constant division among Protestants, few Protestants would know or condone such a statement. Nonetheless, this was so when Luther first took baptism from the Church and remains so until now.
Also worth mentioning is a group known as the "Old Catholics". This group split from the Church when Vatican I made it's declaration on papal infallibility and though rare, when they are encountered, many do not understand that they are not part of the Catholic Church as their name seems to imply.
Within the Church itself are several movements. Some of these movements are probably going to one day become heretical or schismatic, but this remains to be seen. With the Church itself you will find:
Counciliarists - these Catholics believe that a Church Council is more powerful than the pope, a sort of democratic movement that if used must be respected and the pope only acts as its representative. This movement has been increasingly popular since Vatican II. Chances are it will end in heresy.
Liberals - Liberal Catholics are usually not aware they are liberal. The Church seems to have a liberal bias these days and Vatican II is often cited as a victory for the liberals. Liberals seek to make Catholicism more tolerant, to adapt to the world and people and modern views. Much of the constant novelty and change in liturgy and policy is from liberal influence. Everything from the charismatic movement to Eucharistic Ministers is of a liberal flavor. If the Church ever decides things have gone too far, some Liberals may become openly revolutionary and you would probably see a repeat of the Protestant Reformation and a new huge break in the Church. It is for fear of this that many bishops and even Rome do not put their foot down, as the sudden blacklash may cause thousands upon thousands to rebel and leave the Faith.
Conservatives - By definition, Catholics are reserved; they practice a Faith that is unchanging Truth and thus any change is approached with caution and thorough study of the deposit of Faith. Conservatives have become reactionary now because of the swift progress of Liberal forces. Surprised at Vatican II the conservative movement was left scrambling to form. After Vatican II they dissolved for the most part as a definite body and now a Catholic "conservative" is more an attitude than anything else. The conservatives are the ones that dig in their heels to resist change. Because of them many of the novelties the Liberals wish to introduce have been tempered, slowed or blocked. The conservatives wish the Church to settle down and maintain what it has now, many of the current bishops including the pope are basically the first generation of Vatican II revolutionists who have decided that the task is done and they have what they wanted. The second generation, the Liberals, are the new wave of revolutionists, making the first conservative. This is typical of revolution and change for revolutions never stop, they consume themselves "The Revolution eats its own parents."
Traditionalists - Traditionalists are those Catholics who consider Vatican II and its ideas and results so novel that it must not be touched. Because of this, they refuse to change any of the liturgical rites or follow any of the novel policies in pastoral discipline that have been introduced since, usually, 1962. Thus they still celebrate the Tridentine Mass in Latin. The Traditionalist movement is often attributed to Archbishop Marcel Lefebvre of France and Bishop De Castro Meyer of Brazil who together consecrated four bishops in 1988 in order that the Traditional movement would be perpetuated in the event that no bishops of the Church would stay true to the old Traditions. Because of this, the Lefebrve movement, the Society of St. Pius X is branded as schismatic, however, the Church has been giving mixed signals as to whether they actually are. As of today, the majority of canon lawyers when pressed will say they are not, though they certainly must walk a fine line. There have been various Traditional Catholic movements since, but the SSPX continues to be the mainstay.
Sede Vacantists - Sede Vancantism is name given to those Catholics who believe "the see of Peter is vacant". They determine that the changes of Vatican II were so radical that surely the pope committed heresy, and thus, he is no longer pope. There are many different cells of SV, and where and which pope first went wrong varies from as far back as Pius XII to John Paul II. The intrinsic problem with some of these groups is that if the pope has fallen, and the see has been vacant for decades, who is going to be around to elect a new pope? Some have solved this quandry by reasoning - correctly if true - that if all the rest of the Church has plunged into heresy or schism, they are the last Catholics left and thus they elect one of their own number as pope. There are literally dozens upon dozens of such popes around the world since SV groups are fragmented and not united as a solid movement. Some of the more cautious groups are not so radical in their solutions and merely await the Church hierarchy's return to Tradition whereupon things can be set right.
Feenyism - Localized largely to the USA, the Feeneyites are a group that takes its name from Father Leonard Feeney, SJ, deceased 1978. The Feeneyites split at his death but are consistent in that both preach that there is no such thing as "baptism of desire". Although it seems a minor movement, the Feeneyites are spread out and frequently found across the board in Conservative and Traditional parishes, where often they are not welcome because of their adamant stand on this position. Many consider their denial of baptism of desire to be heretical and no doubt the Church will one day have to make a definitive statement if the Feeneyites continue to perpetuate.