The Eastern Orthodox church is comprised of several traditions. Most notable among these are the Greek and Russian. It is the right fork of Christianity's split into two groups (mainly over the issue of the Holy Trinity) in 1050 AD, the other being Catholicism.
Orthodox do not recognize the authority of the Pope, although before the schism, that was the case. The spiritual leader of a given geographic area is a patriarch or metropolitan. The current metropolitan of North America, for instance, is the Most Blessed Herman1. The hierarchy below that resembles the Catholic, from archbishops and bishops to priests and deacons.
The Orthodox Church took a very different approach to church/state relations from the Catholic. It was less of an indepenent power or state than the Vatican, but rather acted as the Byzantine Empire's right hand. There were many conflicts between the secular and clerical authorities, such as the struggle over iconoclasm in 700-900 AD. After the Byzantine Empire collapsed, the various traditions diverged more and more.
Some important Orthodox sites include Mount Athos, the former Hagia Sophia, and the Cathedral of Christ the Saviour in Moscow.
1. There are a number of sects of Orthodoxy in the US, but the largest and arguably most legitimate, to which His Holiness belongs, is the Orthodox Church in America(www.oca.org), formerly a branch of the Moscow Patriarchate. Others include the Synod and COROC, a very conservative church founded and patronized by White emigres from Soviet Russia and their descendants.