The order that's usually given (it's not the only one, but is the one that has largely predominated) is that found in the Dionysian Celestial Hierarchy. The ordering is, from least to greatest (I have also noted what the name is said to denote):
The Order of Angelic Choirs That Are Most Concerned with the Manifest and Mundane
Angel
Archangel
Principality
The Order of Intermediate Angelic Choirs
Virtue: Forceful Strength
Power: Mighty Order
Domininion: Sublime Authority
The Order of Angelic Choirs That Are Immersed in the Presence of God
Throne: Majestic Righteousness
Cherub: Overflowing Wisdom
Seraph: Ardent Love
The names indicate 'interpretive characteristics', i.e., what the choir conveys to every intelligence beneath it. They are, it should be understood, not themselves exclusive; ever label in the above lists applies to every angel who falls in the orders I've listed below it. This is because the whole point of the ordering is that the greater orders possess all the excellences of the lesser ones. Further, lower angels can sometimes be called by the names of higher choirs insofar as they are purified, illuminated, and perfected by means of them. (Likewise, we can sometimes be called angelic to the extent that God has purified, illuminated, and perfected us by means of angels.) However, the names are often used exclusively, so when you say, "Archangel" you usually mean only the archangels in the lowest order.
But simply stating the order simply doesn't do justice to the Dionysian discussion. The whole point, as we find in the first few chapters of the Celestial Hierarchy, is not to discuss angels but to discuss participation in the divine; the orders are levels of participation in divine glory by intelligences, and the reason for discussing it is that we may better understand what participation in the divine involves. For it is our destiny to participate in God's light and love. The alternatives are to talk about this participation in terms of human beings or in terms of God's supereminent possession of excellences. The value of thinking of it in terms of angels rather than humans is that angels are pure cases: we can talk about them without dragging all the finer complications that are involved in talking about human beings. The value of thinking of it in terms of angels rather than God is twofold: (1) God technically doesn't participate God, so it's very difficult (for us, at least) to understand what participating the divine glory is if we only allow ourselves to think about God; (2) and, moreover, we can only think correctly about God's possession of his glory if we bring in the via negativa, whereas we need to think about participation much more directly and positively. So the point of discussing the matter at all is not to satisfy curiosity about angels, but to guide us in union with God:
By moulding itself after their likeness our own hierarchy will, as far as possible, be assimilated to it and will, in very deed, show forth, as in images, the angelic beauty; receiving its form from them, and being uplifted by them to the superessential Source of every Hierarchy.
Indeed, the Dionysian goes so far as to say that the hierarchies are recapitulated, or at least there is something analogous to them, to some degree in every angelic and human intelligence. Moreover, the Dionysian is quite clear that the choirs of angels are symbols of divine providence. As he says, God does not share sovereignty with angels, but through one all-powerful providence he governs through them, lifting all things up to himself by their means. Indeed, this has been the great attraction of the list throughout the ages, that it provides a useful way of thinking about both divine providence and our sanctification. To focus on the angels alone is to miss the point; as the old proverb says, when a finger points at the moon, only a dog looks at the finger.
It's also important to note that the Dionysian is actually rather modest in his claims; for instance, with regard to one question about his ordering he gives the reader (ostensibly Timothy) two possible responses, one of which he favors, and then tells him to pick whichever of the two he thinks more reasonable, or figure out something closer to the truth, or learn it from someone else, and then let everyone know.
It's clear that the Dionysian is philosophically is heavily influenced by Proclus. It's also possible to exaggerate this, however; however influenced he may be, he talks throughout like a Christian. And his attitude here is similar to his attitude in the Divine Names, where he develops his philosophical account of negative theology for the express purpose of driving people to Scripture; the Neoplatonic philosophy is made to recognize the superiority of the Christian life.
