I have traditionally understood orthodox hierarchy in the following way:
-Priests report to their bishop/metropolitan, which in my case would be the Metropolis of Denver.
-The Denver metropolis reports to Archbishop Demetrios in NYC, representing GOA as a whole.
-Archbishop Demetrios reports to Patriarch Bartholomew.
This is the same case for other orthodox churches, such as OCA, Antiochian, etc. I've been told that my understanding of the above listed hierarchy isn't quite correct in that Bartholomew for example is only head of his local following of a couple thousand people in Constantinople. Demetrios is only head of his local area there in NYC and nothing else. Which is correct, or are they both true but different because of religious and/or administrative matters?
Also, what are the exact reasons that define one person to be a Metropolitan (bishop) versus an Archbishop? Or a Patriarch?
This can be a little confusing, because metropolitans, archbishops and patriarchs are all bishops. They are titles of administration and honor, but they are sacramentally equal. Of course, to add to the confusion, these terms are used differently by different traditions (Greek, Slavic, Syriac (Antiochian), etc.) I'll try to explain this. If I err, someone please correct me!
All priests report to their local diocesean bishop, in your case the Metropolitan of Denver of the GOAA. Administratively, above the Metropolitan of Denver is the Archbishop of New York, His Emience Demetrios. Demetrios is the Archbishop of New York and All America because he is the first-among-equals for the GOAA. The first in honor. He speaks on behalf of the entire GOAA and chairs their assembly. However, he has his own diocese, as well. The Diocese of New York. That is his territory. Although he is in honor higher than the other GOAA bishops, they are still all other bishops. He cannot tell them what to do in every circumstance, and he cannot even enter their diocese to serve without permission.
And then, since the GOAA is not an autocephalous church (heading itself), but an arm of the Ecumenical Patriarchate, His All Holiness Bartholomew is "above", in honor, Archbishop Demetrios. Patriarch is a title that is higher in honor than Archbishop or Metropolitan, but he is still a bishop, just as they are. Usually, Patriarchs are heads of autocephalous churches, but not always. Some smaller churches in the EO communion are headed by an Archbishop or a Metropolitan. Side note: In the OO tradition, the ancient title "Catholicos" is preserved as distinct from Patriarch, and is yet another higher title of honor. For example, there are two Patriarchs under the Catholicos of all Armenians.
So, all are bishops, but Metropolitans have higher honor than just Bishops. Archbishops over Metropolitans, and Patriarchs over Archbishops. In the Antiochian/Slavic tradition, there is a reversal. Archbishops in those traditions rank in honor below Metropolitans.
Even among the heads of autocephalous churches, there is a system of honor. The head of an autocephalous church is the first-among-equals for that autocelphalous churches, and an order of honor below him. The same is true among all of the autocephalous churches. The Ecumenical Patriarch is the first-among-equals of all bishops. He is the highest of honor. That said, he is not the universal bishop, as is the Roman Pope for the Latins. He cannot enter any other bishops territory without permission, and has no say in the internal affairs of other churches, unless he is appealed to by them. Below him is an ordering of the autocephalous primates (head bishops), each one of them holding a place of honor. When an autocephalous primate serves the Liturgy, these bishops are commemorated by him at the Great Entrance (where the priest serving would commemorate his bishop).
I hope this has been helpful!