An archbishop presides over a province, a cluster of neighbouring dioceses, and the gap between what that sounds like and what it means is the most persistent confusion in this area.
He does not govern the other dioceses. Their bishops are not his subordinates, he does not appoint or remove them, he cannot overrule their decisions, and he has no routine authority over their finances, clergy or parishes. Reports that an archbishop has ordered something across his province are almost always wrong about the mechanism, even when they are right that something happened.
What he does have is real but narrow: he convenes the province's bishops and presides at their meetings; he has a supervisory function permitting him to investigate where a diocese is manifestly failing and to refer the matter upward; he consecrates new bishops of the province; he has an appellate role in certain canonical cases; and he takes precedence. In the Latin church he receives the pallium, a woollen band conferred from Rome, which marks the relationship to the centre and was for centuries an instrument by which Rome controlled regional appointments, since a metropolitan could not act until he had it.
Anglican provinces give their primates somewhat more convening authority, and the Archbishop of Canterbury's position within the wider Anglican Communion is of that kind: a focus of unity with no jurisdiction outside his own province, which is why the Communion's divisions of the last twenty years have had no mechanism for resolution.
Orthodox usage is more variable still. Metropolitan, archbishop, exarch and patriarch carry different weights in different autocephalous churches, and in some of them an archbishop outranks a metropolitan while in others the reverse holds. The titles are not a single system.