Hierarchy means 'priestly rule', but could also be understood as 'holy order'. One of the most influential accounts of the latter is that found in the Dionysian corpus, particularly in the Celestial Hierarchy and the Ecclesiastical Hierarchy.
The central notion of the Dionysian account of hierarchy is found in part in James 1:17; holy order is the structure of an exitus and a reditus, arising out of a procession of good from God to God. Both the celestial hierarchy (the angelic orders) and the ecclesiastical hierarchy (which is a symbolic copy of the celestial hierarchy) exemplify this. It is in the celestial hierarchy in particular that we find the proper definition of 'hierarchy' fulfilled (Celestial 3): "Hierarchy is, in my opinion, a holy order and knowledge and activity which, so far as is attainable, participates in the Divine Likeness, and is lifted up to the illuminations given it from God, and correspondingly towards the imitation of God." Thus hierarchy is an uplifting. The way in which it does is through rites, which here are understood as being acts in the hierarchy. We can therefore understand hierarchy in this way, as well, through the kinds of acts constituting it and to which it in turn gives contextual meaning: "Every Hierarchy, then, is, according to our august tradition, the whole account of the sacred things falling under it, a most complete summary of the sacred rites of this or that Hierarchy, as the case may be" (Ecclesiastical 1).
Given the aim and purpose of hierarchy, then, which is imitation of and union with God, every intelligence allotted a place in the hierarchy is lifted up toward God. But given the nature of intelligence, as that intelligence becomes more like God, it expresses this divinelikeness to others, and through that expression those others may also be lifted up toward God. Hierarchy, of course, is not an individual imitation but a social cooperation in progress in which one person helps another. But this helping is also not individual; it is essential to the notion that God helps people both directly and through other people, so that when someone in the hierarchy expresses their divinelikeness in such a way as to help another, this is actually a cooperation with God.
The entire hierarchy, in fact, is constituted as a system of these interactive cooperations (Celestial 3): "For the holy constitution of the Hierarchy ordains that some are purified, others purify; some are enlightened, others enlighten; some are perfected, others make perfect; for in this way the divine imitation will fit each one." These are the three hierarchical acts: purification, illumination, and completion or perfection. They can be considered three different components that go into deification or divinization, which, again, is the purpose of hierarchy, making people more like God. Purification makes one fit for higher union with God; illumination gives the essential elements for this, such as knowledge or virtue; and perfection completes the process in the actual attainment of that higher union. These hierarchical acts have a number of important characteristics that must be kept in mind to understand the functioning of hierarchy:
(1) They are directional. That is, they are always from one person to another, and this from-to direction is itself a part of a larger exitus (procession from God in creation) and reditus (return to God in union).
(2) They are cooperative on the part of the higher. The from-ward or higher person in the hierarchical act has some kind of greater union with God that is being communicated to the to-ward or lower person in the act; divinelikeness flows down through them. This requires first, the cooperation of the lower, and second, divine cooperation.
(3) They are cooperative on the part of the lower. The to-ward or lower person in the hierarchical act is achieving a greater degree of fulfillment, which is union with God; this is something that can only be had in cooperation, both with the higher person and with God.
(4) They can be reiterated at many different levels. There are many different gradations in union with God, and therefore having achieved perfection in one may open up ways to be purified for another.
The purest form of this is that found in the angelic hierarchy. Angels are the intelligences closest around God, and are given that name because they preeminently receive and pass on divine things; their whole office and function is communication of closeness to God in one way or another. Contrary to many sarcastic Protestant and secular comments on the Dionysian hierarchy of angels, the Dionysian author is extremely clear that we do not know much at all about the angels. He regards the angelic or celestial orders to be in some sense beyond our capacity to understand. This would be the end of it, except for Scripture. (It is absolutely essential to understanding the Dionysian corpus that it is a theology of Scripture; it is always and everywhere about the Scripture. This is very explicit, but is very often forgotten in focusing on the Neoplatonic vocabulary that the author derives from Proclus.) Scripture, as divine revelation gives us information about angels. It gives us very little, and almost entirely in symbolic terms. But its whole point as divine revelation is to raise our minds to divine truths, and this is true of its symbols, as well. By these symbols Scripture can raise our minds to genuine truths about these orders. That the symbols are strange, discordant, and diverse increases, rather than decreases, their value, because they serve to remind us of how little we know.
Thus the Dionysian author explicitly says that we do not know for certain how the angels are organized in their hierarchy. What we do have, however, are indications from Scripture, in the form of 'interpretive names'. These are of various different kinds, but some of these names can be understood in such a way as to associate them with the hierarchical acts. This combination of a general conception of hierarchy and symbolic names gives us the famous Dionysian orders of angels, a set of three tiers of three orders each, each tier being associated with a hierarchical act, and each order within each tier also being associated with a hierarchical act. Each order in turn has a godlike characteristic associated with both its particular hierarchical act and its interpretive name:
perfecting perfecting -- seraphim -- love
perfecting illuminating -- cherubim -- knowledge
perfecting purifying -- thrones -- justice
illuminating perfecting -- dominions -- free lordship and authority
illuminating illuminating -- virtues -- irresistible force
illuminating purifying -- powers -- regulative order
purifying perfecting -- principalities -- leadership
purifying illuminating -- archangels -- interpretation
purifying purifying -- angels -- care for mundane things
Here we see the iterability of the hierarchical acts creating levels of union with God, from angels, the lowest part of the celestial hierarchy, to seraphim, the highest. Each of these names comes from Scripture, and is interpreted etymologically. Thus, for instance, 'seraphim' means 'fiery ones', so is associated with the purest fire of love, the most perfecting of all perfecting acts. Each tier in some sense imitates the higher tiers, so that, for instance, as we move from care for mundane things to regulative order to justice, or from leadership to lordship to love, we get a more and more pure form of the godlike characteristic that constitutes that kind of union with God, arising from the higher version of that hierarchical act.
All of the godlike characteristics are features of divine providence, with love, knowledge, and justice being the most pure cases. The Dionysian hierarchy is thus a picture of divine providence; all the works of divine providence express, in one way or another, these godlike characteristics. But it's not a mere picture; the angelic hierarchy is itself an expression and instrument of divine providence. The angels exhibit the features of providence because they are providential ministers, and they are providential ministers by virtue of their place in the hierarchy, i.e., by virtue of their hierarchical acts. And, of course, they are ministerial because all hierarchical acts are cooperative. "Each Order is the interpreter and herald of those above it, the most venerable being the interpreter of God who inspires them, and the others in turn of those inspired by God." (Celestial 10). Each is actually cooperating with the higher orders, all the way up to God Himself, in their own particular work, and likewise the higher work through the lower.
The third tier is the order concerned with purifying, enlightening, and perfecting what is beneath the angels -- namely, human beings. Thus hierarchy, sacred order, does not end at the angels but is communicated downward by them into a human hierarchy. The Dionysian is clear that (despite its being human) we do not wholly understand this hierarchy, either, precisely because it is constituted by cooperation with all of the hierarchical orders above it, up to God Himself. As with the angels, we understand it symbolically, in this case in the liturgy and the sacraments, which are also the means whereby human beings purify, illuminate, and perfect each other. (The Dionysian discusses a wide but somewhat mixed and non-exhaustive assortment of these; the principle of selection seems to be a mix of chronological, from baptism to funeral, and the liturgies that give the clearest information on hierarchical acts.) Of these, the Eucharist is the highest of the perfecting acts, assimilating us to God. We human beings cooperate with the angels, and (through and in and with the angels) with God through liturgical and sacramental practice. This gives us the human hierarchy, which is 'flatter' than the more perfect angelic hierarchy, but which serves as a symbolic copy of it:
perfecting -- bishops (hierarchs)
illuminating -- priests
purifying -- deacons (leitourgoi)
perfecting -- monastics (i.e., consecrated laypersons)
illuminating -- contemplatives
purifying -- multitudes
The ecclesiastical hierarchy, then, is not coextensive with the clergy but with the whole Church; to participate in the Church is to participate in the hierarchy that descends from God. What we call the laity are a key part of the hierarchy; they are, so to speak, the principalities, archangels and angels of humanity, and their task is to purify, illuminate, and perfect the world. Because of how hierarchy works, the lower orders cannot be treated as unholy -- after all, everything in a hierarchy is sacred, and people insofar as they are participate in it are, in fact, holy, none excepted. Each is in fact genuinely united to, and expressive of, God in its own way, although the higher orders in a more godlike fashion. You can fall out of the hierarchy into rebellion (like the fallen angels with the angelic orders), but within the sacred order your work is sacred, part of the movement from God to God.