A church sui iuris is "a community of the Christian faithful, which is joined together by a hierarchy according to the norm of law and which is expressly or tacitly recognized as sui iuris by the supreme authority of the Church" (CCEO.27). The term sui iuris is an innovation of the CCEO, and it denotes the relative autonomy of the oriental Catholic Churches. This canonical term, pregnant with many juridical nuances, indicates the God-given mission of the Oriental Catholic Churches to keep up their patrimonial autonomous nature. And the autonomy of these churches is relative in the sense that it is under the supreme authority of the Roman Pontiff.
The Latin sui iuris (the individual words meaning 'self' and 'law') corresponds to the Greek '', from which the English word autonomy is derived.
Church documents such as the Code of Canons of the Eastern Churches apply the Latin term sui iuris to the particular Churches that are together the Catholic Church, the Roman Catholic Church and those in communion with it. By far the largest of the sui iuris churches is the Latin Church or the Latin Rite. Over that particular church, the Pope exercises his papal authority, and the authority that in other particular churches belongs to a Patriarch. He has, therefore, been referred to also as Patriarch of the West. The other particular Churches are called Eastern Catholic Churches, each of which, if large enough, has its own patriarch or other chief hierarch, with authority over all the bishops of that particular Church or rite.
"The Eastern Catholic Churches are not 'experimental' or 'provisional' communities; these are sui iuris Churches; One, Holy, Catholic and Apostolic, with the firm canonical base of the Code of Canons of the Eastern Churches promulgated by Pope John Paul II." 
"It would likewise be helpful to prepare an Empathetical Directory that would 'take into account the special character of the Eastern Churches, so that the biblical and liturgical emphasis as well as the traditions of each Church Sui Iuris in petrology, hagiography and even iconography are highlighted in conveying the catechesis' (CCEO, can. 621, §2)" John Paul II
"On behalf of the Kyrgyzstan Catholics I would like to express our gratitude to the Holy Father (i.e., the Pope) for his prayers and for all that he has done for us: ... and for the creation of the new 'missioni sui iuris' in Central Asia, in a special way — for the trust placed on the 'Minima Societas Jesu', to which he entrusted the mission in Kyrgyzstan." 
"...[T]he rays originating in the one Lord, the sun of justice which illumines every man (cf. Jn1:9), ... received by each individual Church sui iuris, has value and infinite dynamism and constitutes a part of the universal heritage of the Church." "Instruction for Applying the Liturgical Prescriptions of the Code of Canons of the Eastern Churches", issued January 6, 1996 by the Congregation for the Eastern Churches .
Categories of sui iuris churches
According to CCEO, the Oriental Catholic churches sui iuris are of four categories.
A patriarchal church is a full-grown form of an Eastern Catholic church. It is 'a community of the Christian faithful joined together by' a Patriarchal hierarchy. The Patriarch together with the synod of bishops has the legislative, judicial and administrative powers within jurisdictional territory of the patriarchal church, without prejudice to those powers reserved, in the common law to the Roman pontiff (CCEO 55-150). Among the catholic oriental churches the following churches are of patriarchal status:
Melkite Greek Catholic Church (definitively 1726): Damascus, (1,346,635), Syria, Lebanon, Jordan, Israel, Jerusalem, Brazil, United States, Canada, Mexico, Iraq, Egypt and Sudan, Kuwait, Australia, Venezuela, Argentina
Major archiepiscopal churches
Major archiepiscopal churches are the oriental churches, governed by the major archbishops being assisted by the respective synod of bishops. These churches also have almost the same rights and obligations of Patriarchal Churches. A major archbishop is the metropolitan of a see determined or recognized by the Supreme authority of the Church, who presides over an entire Eastern Church sui iuris that is not distinguished with the patriarchal title. What is stated in common law concerning patriarchal Churches or patriarchs is understood to be applicable to major archiepiscopal churches or major archbishops, unless the common law expressly provides otherwise or it is evident from the nature of the matter" (CCEO.151, 152). Following are the Major Archiepiscopal Churches:
A sui iuris church which is governed by a metropolitan is called a metropolitan church sui iuris. "A Metropolitan Church sui iuris is presided over by the Metropolitan of a determined see who has been appointed by the Roman Pontiff and is assisted by a council of hierarchs according to the norm of law" (CCEO. 155§1). The Catholic metropolitan churches are the following:
Other than the above-mentioned three forms of sui iuris churches there are some other sui iuris ecclesiastical communities. It is "a Church sui iuris which is neither patriarchal nor major archiepiscopal nor Metropolitan, and is entrusted to a hierarch who presides over it in accordance with the norm of common law and the particular law established by the Roman Pontiff" (CCEO. 174). The following churches are of this juridical status:
Russian Greek Catholic Church (1905) - two apostolic exarchates, at present with no published hierarchs: Russia, China; currently about 20 parishes and communities scattered around the world, including five in Russia itself, answering to bishops of other jurisdictions
In civil law, the phrase sui juris indicates legal competence - the capacity to manage one's own affairs (Black's Law Dictionary, Oxford English Dictionary) -- as opposed to alieni juris, which means someone under the control of another (such as a child or mentally incapable person might be).
It also indicates a person capable of suing and/or being sued in a legal proceeding in his own name (in personam) without the need of an ad litem, that is, a court appointed representative, acting on behalf of a defendant, who is deemed to be incapable of representing himself.
That is in contrast with many parliamentary democracies, like Canada and the United Kingdom, where the Queen (the head of state), at the request of the Prime Minister (the head of government), has power to convene, prorogue, or dissolve Parliament, which has no choice in the matter. (However, in 2019 the United Kingdom Supreme Court ruled that the Prime Minister's advice to HM The Queen to prorogue the UK Parliament was designed to deprive Parliament of the opportunity to debate matters relating to the UK's leaving the European Union and was therefore unlawful and that the prorogue never, therefore, happened.)
^"Collins English Dictionary". HarperCollins Publishers. 2003. Retrieved 2012. sui juris ['su:a? 'dr?s] adj (Law) (usually postpositive) Law of full age and not under disability; legally competent to manage one's own affairs; independent [from Latin, literally: of one's own right]
^Original italian: "Una Chiesa Orientale cattolica è una parte della Chiesa Universale che vive la fede in modo corrispondente ad una delle cinque grandi tradizioni orientali- Alessandrina, Antiochena, Costantinopolitina, Caldea, Armena- e che contiene o è almeno capace di contenere, come sue componenti minori, più comunità diocesane gerarchicamente riunite sotto la guida di un capo comune legittimamente eletto e in comunione con Roma, il quale con il proprio Sinodo costituisce la superiore istanza per tutti gli affari di carattere amministrativo, legislativo e giudiziario delle stesse Communità, nell'ambito del diritto comune a tutte le Chiese, determinato nei Canoni sanciti dai Concili Ecumenici o del Romano Pontefice, sempre preservando il diritto di quest'ultimo di intervenire nei singoli casi". pp. 103-104.