Hierarchy is something you are familiar with based on the transcendental perfection of the natural world. Transcendental means from the bottom up. Matter is at the lowest level of perfection. Above it are plants which used the soil, water and the energy of the sun to grow. Still higher are animals which can move and have sense perception and the ability to decide what is good or bad for them. Above all is man with his intelligence and will which can rule all the lower creatures.
The angelic hierarchy is based on the degree of participation in the beatific vision of each angel, on the natural gifts God has given each such as intelligence, power, wisdom and other supernatural gifts. Unlike the order in nature, this is a holy and descending order because it flows from God downward. If this was the case there would be only one hierarchy under God ruling each directly. However, this is not the case because God delegates authority and power to different angels who have been grouped into three ruling hierarchies each containing three orders of angels.
Basing himself on the Bible, "Dionysius the Areopagite from Acts 17:34... wrote treatises on a Christian Celestial Hierarchy (dealing with realms of angels and angelic beings), the Divine Names of God, and certain aspects of Mystical Theology. Scholars do not think he wrote the treatise because he depended on the philosophy of Proclus, a Neo-platonist, who died c. 485 CE. So, Pseudo Dionysius likely produced in work in the sixth century. (Wikipedia: Engaging Pseudo-Dionysius)
Pseudo Dionysius reduced each angelic hierarchy to tree orders. Why? St. Thomas Aquinas gives a clear explanation why this is appropriate in the (Summa, Pt. I, Q. 108, Art.2) First, one hierarchy is one principality under a prince. The three orders are organizations which hold certain offices and actions. There are three orders because like in human society there are those who judge, those who fights and the majority who do the work. Without orders the prince or angel in charge, would not be able to rule the multitude of angels in the hierarchy.
What follows is a short description of the work or each angelic hierarchy:
Seraphims are angels on fire with the love of God who continually praise him as Isaiah tells us, "One cried out to the other: Holy, holy, holy is the LORD of hosts! All the earth is filled with his glory! (Is 6:2) In Isaiah's vision he realized that he was a sinful man, so "One of the seraphim flew to me, holding an ember which he had taken with tongs from the altar. He touched my mouth with it. See, he said, now that this has touched your lips, your wickedness is removed, your sin purged. (Is 6:6-7)
Cherubims signify "fullness of knowledge" and were sent by God to expel Adam and Eve from the Garden of Eden as we read, "He expelled the man, stationing the cherubim and the fiery revolving sword east of the garden of Eden, to guard the way to the tree of life." (Gin 3:24) Moses was directed to place two Cherubims on top of the Ark of the Covenant. Thrones are involved in carrying out justice in the heavens, that is, to carry out the justice of God.
Dominions have the task of relegating duties to lower angels. They preside over nations. Virtues are responsible for maintaining the physical universe as explained on page 2. Powers sometimes serve as warren angels fiercely loyal to God. They work to oversee the distribution of power among mankind.
Principalities have the duty to carry out the orders given to them by Dominions. The inspire those working in branches of knowledge. Archangels govern affairs of the messengers of various realms. We read, "Then war broke out in heaven; Michael and his angels battled against the dragon. The dragon and its angels fought back, but they did not prevail and there was no longer any place for them in heaven." (Rv 12:7) Michael is an archangel, a leading angel of the angelic realm. Angels means messenger and they commune with various life forms in the physical universe and these include guarding angels. Christ specifically pointed out guardian angels when he said, See that you do not despise one of these little ones, for I say to you that their angels in heaven always look upon the face of my heavenly Father." (Mt 18:10)
An observation is here in order: The hierarchy of angels is a work of reason based on revelation which has inspired many poets, artist and philosophers. It falls far short of the reality but it is a worthy subject to Contemplate.