According to the list on the US Conference of Catholic Bishops, there are currently 15 Eastern Rite Eparchies (dioceses) and 2 Arch-eparchies (archdiocese) in the USA. I was surprised there were this many, as I was familiar with only a few of them by name.
I have endeavored to add links to the official websites of all such eparchies as I could find. There were a few that appeared to have no office website, as far as I (or more precisely Google) could find. Each of them has some interesting information to peruse. It is worth it just to go and look at the various ecclesiastical vestments of the various bishops. They know snappy vestments in the East.
They are as follows:
Eastern Rite Eparchies
1. St. Thomas Syro-Malabar Diocese of Chicago (Illinois)
2. Eparchy of St. George in Canton for the Romanians (Ohio)
3. Ukrainian Catholic Eparchy of Stamford (Connecticut)
4. Byzantine Catholic Eparchy of Passaic (New Jersey)
5. Holy Protection of Mary Byzantine Catholic Eparchy of Phoenix (Arizona)
6. Eparchy of Our Lady of Deliverance Syriac Catholic Diocese in the U.S. and Canada (New Jersey)
7. Chaldean Eparchy of St. Thomas the Apostle of Detroit (Michigan)
8. Chaldean Eparchy of St. Peter the Apostle, San Diego (California)
9. Eparchy of Parma, Ohio (Ruthenian)
10. Eparchy of St. Maron of Brooklyn (New York)
11. Eparchy of Our Lady of Nareg in New York of Armenian Catholics
12. Eparchy of Newton, Our Lady of the Annunciation, Boston (Melkite Greek)(Massachusetts)
13. Eparchy of St. Nicholas in Chicago for Ukrainians (Illinois)
14. Eparchy of Our Lady of Lebanon (Maronite)
15. Ukrainian Catholic Eparchy of St. Josaphat, Parma (Ohio)
Eastern Rite Archeparchies
Ukrainian Catholic Archeparchy of Philadelphia (Pennsylvania)
Byzantine Catholic Archeparchy of Pittsburgh (Pennsylvania)
Also noted on the USCCB website were the following statements:
"Eastern Catholic Churches are churches with origins in Eastern Europe, Asia and Africa that have their own distinctive liturgical, legal and organizational systems and are identified by the national or ethnic character of their region of origin. Each is considered fully equal to the Latin tradition within the church. In the United States, there are 15 Eastern church dioceses (called eparchies) and two Eastern church archdioceses (or archeparchies), the Byzantine Catholic Archdiocese of Pittsburgh and the Ukrainian Catholic Archdiocese of Philadelphia.
"The apostolic exarchate for the Syro-Malankara Catholic Church in the United States is headed by a bishop who is a member of the U.S. Conference of Catholic Bishops. An apostolic exarchate is the Eastern Catholic Church equivalent of an apostolic vicariate. It is not a full-fledged diocese/eparchy, but is established by the Holy See for the pastoral care of Eastern Catholics in an area outside the territory of the Eastern Catholic Church to which they belong. It is headed by a bishop or a priest with the title of exarch."