"When I think of the Orthodox Church, I think of the Greek Orthodox Church - is that what he is referring to?"
The Greek Orthodox Church is one of the Orthodox Churches. There are also Russian, Serbia, Antiochian and Romanian Orthodox Churches. They are pretty much all in communion with one another, and someone who has been baptized in one is generally accepted by all of them.
They are run by autocephelous patriarchs, who are elected by the upper clergy. The patriachs are coequal, and church policy can only be changed at ecumenical councils with representatives from all the Orthodox Churchs. In reality the last one was held in the 6th or 7th century, and the churches pretty much go their own ways.
There is also an autocephalous church here, the
Orthodox Church in America of which I am a member.
In fact, there are fifteen canonical Orthodox Churches
around the world and may others that come under the jurisdiction
of one of those fifteen (e.g. Finland, Japan, Uganda, etc.)
We all share the same faith but different languages and cultures.