Thought I’d answer your questions, too, not because other posters - Catholic and Orthodox - haven’t done a great job, but because our answers will complement each other. I have a few extra details for you.
Ok, can someone help me understand these things?
Is Eastern Catholic different from Orthodox?
Yes. Many (
though far from all) eastern Catholics have the same
rite as the Orthodox - the Byzantine Rite (a “rite” is the “liturgical, theological, spiritual and disciplinary patrimony, culture and circumstances of history of a distinct people, by which its own manner of living the faith is manifested in each autonomous [sui iuris] Church” according to the
Code of Canons for the Eastern Churches).
But the Orthodox Church is not in communion with the eastern Catholic churches, which are a part of the Catholic Church.
Is it Orthodox or Orthodox Catholic?
Orthodox Christians often refer to themselves as “Orthodox Catholics,” because the official title for their church is “the Orthodox Catholic Church.” They call themselves this because they consider their church to be the one, holy, catholic, and apostolic Church founded by Christ.
Obviously the Catholic Church claims this, too, so generally Catholics refer to themselves as “Catholic” and refer to the Orthodox simply as “Orthodox.”
Further complicating matters is that many Catholics who want to communicate their fidelity to the Church’s teachings refer to themselves as “orthodox Catholics.” In this context the term is meant to contrast with “heterodox Catholic,” and it is not to be confused with the Orthodox self-label “Orthodox Catholic.”
Am I right in thinking that all Eastern churches are in communion with Rome but Orthodox is not?
You’re partially right: the Orthodox are
not in communion with Rome. The eastern Catholic churches
are, but there are many different kinds of eastern Christians:
(1)
the churches of the Orthodox communion -
not in communion with Rome since roughly the eleventh/twelfth centuries
(2)
the churches of the Oriental Orthodox communion -
not in communion with Rome or with the eastern Orthodox since A.D. 451
(3) the
Assyrian Church of the East and the
Ancient Church of the East, out of communion with everyone since A.D. 431; the latter split from the former in the 1960s over calendar issues (I think)
(4)
the eastern churches that are a part of the Catholic communion -
are in communion with Rome; they re-entered the Catholic Church from the above churches at different points in Church history. Exceptions who were never out of communion with Rome include the Maronite Syriac Church of Antioch, the Italo-Albanian (Byzantine) Catholic Church, and maybe the Syro-Malabar Catholic Church? I’m not sure about that last one; they have a very complicated history.
What are some names of “Eastern” churches? How do I know if they are in communion with Rome? I know the Byzantine church is one, but that is all.
“Byzantine” is a
rite, not a
church. No less than fourteen autonomous Catholic churches share the Byzantine Rite.
Wikipedia lists each of the twenty-two autonomous eastern Catholic churches by name
here.
If you mean “how do I identify an eastern Christian parish as Catholic/in communion with Rome?”, well, usually they’ll have “Catholic” in their name. This is not foolproof, though.
Are Greek Orthodox and Russian Orthodox the same church? Are they in communion with each other?
They are different autonomous churches, but they
are in communion with each other just as the Latin Church is in communion with the eastern Catholic churches. They are a part of the larger body known as the Orthodox Church (the Byzantine one I mentioned above as no longer in communion with Rome since the eleventh/twelfth centuries).
Do Eastern Catholics and Orthodox have Divine Liturgies and how is that different from Roman Mass?
Yes, they do!

There are lots of differences because the Byzantine Rite is a separate tradition from the Roman Rite, but both are 100% valid, beautiful Eucharistic Liturgies.
Can I receive communion at Eastern or Orthodox churches if I am Roman Catholic?
At an eastern Catholic church,
yes.
At an Orthodox church,
no. The Orthodox do not generally permit non-Orthodox to receive Holy Communion at their Liturgy.
Hope this helps!