A Church sui iuris is a Catholic church with its own hierarchy in full communion with the reigning pope as explained by Aramis above. Thus the Latin (or Roman—here in the U.S., we are the majority), Maronite, Ukrainian Greek Catholic, Ruthenian Catholic, Coptic Catholic, Chaldean Catholic, etc., etc. Each is governed by either a patriarch, a major archbishop, a metropolitan archbishop, or some other hierarch.
To make a canonical transfer, you have to have been practicing your Catholic faith in the Church desired for a minimal amount of time—I think 1-2 years is the bare minimum—I recommend more, since your are making a once in a lifetime change. This is important, since,as noted above, with some exceptions, you can only change Churches once in your lifetime and there is some variation in the canon law (an Eastern Catholic cannot marry his/her sibling in law after the death of his/her spouse, for example, or for that matter, a person whom they sponsored as a baptismal godparent, w/o an idult from the Holy See). Once you have been practicing your faith in the desired Church for a while, you petition the local diocesan/eparchial bishop of the Church you want to go to, who then receives the assent of the one you are leaving. He then writes to the Congregation for the Eastern Churches, who acts in the name of the reigning pope and decides whether to grant your request. That’s the basics of it. (Or maybe you petition the Congregation directly, who then receives the assent of the bishops. I can’t remember.) This is assuming you are already a practicing Catholic. For those who are not, canon law dictates that when you become a Catholic, you automatically become a member of the Church which corresponds to the Church or community you are coming from regardless of where you are received into full communion with the Catholic Church (say, a Ukrainian Orthodox Christian would become a Ukrainian Catholic, for example—don’t quote me on this, consult a canon lawyer if you’re in this situation). You could then initiate a petition to transfer.
That being said, don’t just change Churches just b/c you are interested in becoming a married priest (I’m not saying that you are). You have to be sure that you are happy worshipping in the Church of interest and can live with the laws and/or customs of said Church (Eastern Catholic have different fasting rules before Divine Liturgy, I believe, and they also don’t have confessionals, with the exception of the Maronites (?)----also, with the exception of the Maronites, the Eastern Churches don’t use instruments in divine worship). That being said, I highly encourage my fellow Latin Catholics to visit an Eastern Catholic DL at least once in their lifetime. And similarly, I would like to encourage my brethren in the Eastern Churches to visit an AU Mass or a TLM (I haven’t been to a TLM myself) sometime. But I can’t post this last invite here since we’re in the Eastern forums, and it would be off-topic.
I mean once a person is a Catholic they dont have to label themselves anything other than ‘‘Catholic’’ when attending Eastern rite mass on a regular basis right?
I was looking for information on them a month ago.