If you want to understand how the Orthodox Church differs from the Catholic Church on administration both at the local and "universal" levels, the below article is a pretty good gloss of the topic.
(mospat.ru) - The problem of primacy in the Universal Church has been repeatedly raised during the work of the Joint International Commission on Theological Dialogue Between the Orthodox Church and the Roman Catholic Church. On March 27, 2007, the Holy Synod of the Russian Orthodox Church instructed the Synodal Theological Commission to study this problem and draft an official position of the Moscow Patriarchate on the problem (Minutes, No. 26). Meanwhile, the Joint Commission at its meeting on October 13, 2007, in Ravenna, working in the absence of a delegation of the Russian Church and without consideration for her opinion, adopted a document on the Ecclesiological and Canonical Consequences of the Sacramental Nature of the Church. Having studied the Ravenna document, the Russian Orthodox Church disagreed with it in the part that refers to synodality and primacy on the level of the Universal Church. Since the Ravenna document makes a distinction between three levels of church administration, namely, local, regional and universal, the following position taken by the Moscow Patriarchate on the problem of primacy in the Universal Church deals with this problem on the three levels as well.Complete article here.
In the Holy Church of Christ, primacy belongs to her Head – our Lord and Saviour Jesus Christ, the Son of God and the Son of Man. According to St. Paul, the Lord Jesus Christ is the head of the body, the church: who is the beginning, the firstborn from the dead; that in all things he might have the pre-eminence (Col. 1:18).
According to the apostolic teaching, the God of our Lord Jesus Christ, the Father of glory, raised him from the dead, and set him at his own right hand in the heavenly places, far above all principality, and power, and might, and dominion, and every name that is named, not only in this world, but also in that which is to come.And hath put all things under his feet, and gave him to be the head over all things to the church,which is his body (Eph. 1:17-23).
The Church, which is on the earth, represents not only a community of those who believe in Christ but also a divine-human organism: Now ye are the body of Christ, and members in particular (1 Cor. 12:27).
Accordingly, various forms of primacy in the Church in her historical journey in this world are secondary versus the eternal primacy of Christ as Head of the Church by whom God the Father reconciles all things unto himself, whether they be things in earth, or things in heaven (Col. 1:20). Primacy in the Church should be in the first place a ministry of reconciliation with the aim to build harmony, according to the apostle who calls to keep the unity of the Spirit in the bond of peace (Eph. 4:3).
In the life of the Church of Christ, which lives in this age, primacy, along with synodality, is one of the fundamental principles of her order. On various levels of church life, the historically established primacy has a different nature and different sources. These levels are 1) the diocese (eparchy), 2) the autocephalous Local Church, and 3) Universal Church...