Although there are more than 20 other Catholic churches (or rites, as we used to call them) because Eastern Catholics are a small number compared to Western Latin Catholics, we often know little about them or we confuse them with the Orthodox. I can't do justice to this topic in my response but I will try to give briefly some of what I think is important.
An historical look
To even begin to understand them, we need to get a quick glimpse of history. Although early Christian churches developed different customs often according to the area in which they arose, they remained united.
When differences of belief arose, usually due to some political or religious authority deviating from what was regarded as orthodox doctrine, Church councils were called to discuss the issue. If the person or group refused to accept the decision of the council, they were usually condemned as heretics.
Therefore, although a variety of different ways of celebrating Christianity were accepted, the core beliefs remained intact.
The definitive split between Eastern and Western Christianity developed gradually, taking about 1,000 years. Differences grew due to great distances separating them, especially in times of war and turmoil but also due to language (Eastern Greek and Western Latin), culture, mentality and politics.
For the East, ritual was faith in action and so changing the ritual was changing the faith.
For the East, the pope held only a primacy of honour but the papacy, as successor to Peter, claimed the power to intervene in the universal Church. Christianity in the West developed along the Roman model of law and government. Therefore, it tended to be practical and orderly, not mystical or reflective, as was the Eastern way of being Church.
By 1054, after a number of attempts at greater understanding had failed, there seemed to be a genuine desire for reconciliation. However, the cardinal sent by Rome was quite inflexible as was the patriarch of Constantinople.
Old grievances quickly surfaced and instead of reconciliation, the result was excommunication of one another.
This excommunication was only lifted in 1965 by a joint declaration of Pope Paul VI and the Patriarch Athenagoras. Efforts at reconciliation continued under the leadership of Pope John Paul II.
It is interesting to note that the Eastern Church called itself Orthodox, maintaining it was faithful to the Church founded by Christ while the Western Church believed the same about itself, hence catholic or universal.
Orthodox and Catholic churches, although different in liturgical practices, both maintain the sacraments and the ordained priesthood. Catholics accept the validity of Orthodox ordination and therefore, their sacraments.
Gradually, portions of these various Eastern Orthodox churches sought full communion with Rome and became Eastern Catholic churches. Most of the Eastern churches have Catholic and Orthodox counterparts, titled by their origins with language, worship, liturgical and extra-liturgical practices being much the same.
The Eastern Catholic churches are fully Catholic. Pope John Paul II, in his 1995 apostolic letter Orientale Lumen, said that their union with Rome must not imply a diminished awareness of their authenticity and originality.
As Vatican II encouraged them to re-discover their full identity, the pope also stated that they have the right and duty to govern themselves according to their special traditions.
Pope John Paul II maintained that the Church must learn to breathe with both its lungs - the East and the West.
The Catholic Church, in its unity, has a rich and colourful rainbow of liturgies and practices. We, in the Western Church, need to appreciate our Eastern sisters and brothers and learn from them, especially from their rich traditions and spirituality.