This response has been extracted from a longer article of mine on the subject (linked at the end) , hence the discontinuous numbering of its sections.
1. Ousía in Greek philosophical discourse
Deriving then from the verb "to be", as we can see in  ousía enters the Greek philosophical discourse with Plato, who uses it to mean the primary, fundamental kind of being, ("prōtē ousíā", pl. "prôtai ousíai"), with Aristotle afterwards using it with the same meaning. It is crucial to note that, for both Plato and Aristotle, ousía does not enter the discourse as a definition but as a question to be answered: what is the primary, fundamental kind of being? What is prōtē ousíā?
While ousía means the same for both Plato and Aristotle as a question, they provide completely different answers to that question. Plato's answer is well known: the Form, which exists in a transcendent world of Forms, is prōtē ousíā, the primary, real being, of which the particular objects in the sensible world are just shadows. Against this Platonic background, Aristotle then provides his own answer in two works, Categories and Metaphysics, in which, notably, he provides different answers.
Categories, part 5 
Primary ousía is the individual, particular, concrete entity, of which species and accidents can be predicated without it being predicable of or attributable to anything else. This is actually stated in negative terms: “what is neither in a subject nor said of a subject”, i.e. the particular subject itself or "hypokeimenon" , literally “that which underlies or lies beneath” the universals (first of all species and genus) in which it falls and the accidents which inhere in it.
Secondary ousía ("deutérā ousíā", pl. "deúterai ousíai") is the species (first of all) and the genus to which the particular subject belongs.
Metaphysics, book VII/Zeta 
Primary ousía is the essence of the particular entity, which is its form, while the particular entity, the composite of form and matter, is ousía in a derivative sense. (The Aristotelian expression that the Latins translated as "essentia" is "to ti ên einai", "the what it was to be", although sometimes he uses the shorter expression "to ti esti", "the what it is".)
Species and genus are not ousía.
A question arises at this point: Is for Aristotle the form of a particular entity a particular or a universal? This, in conjunction with his statement in Z.13 that no universal is ousía, is the most disputed issue regarding Aristotle's Metaphysics, and has given rise to a whole field of Aristotelian exegesis, in which the main lines are  :
Forms are not universal, and each particular entity has its own form which resides in that entity, so that all individuals of a given species have forms which are identical to one another but numerically different.
Forms are universal, and Z.13 actually does not exclude them as ousía.
Since the second line is basically a reversion to Platonism, IMO it is extremely implausible that it may reflect Aristotle's personal position.
Summarizing the answers that Aristotle gives to the question "What is primary ousía?":
Categories: the particular entity or "hypokeimenon", the subject which underlies all predicates and cannot be predicated of anything else.
Metaphysics: the essence of the particular entity, which is its form, which may be understood in a particular or a universal sense.
2. Hypostasis, the original cognate of substantia
Etymologically  ,
hypostasis = hypó ("under") + stásis ("a standing" = (hístēmi ("to stand") + -sis, verbal noun suffix)) = "that which stands under"
is a direct cognate of    :
substantia = sub ("under") + stans ("standing", present active participle of stō ("stand")) = "that which stands under".
According to , the first recorded use of hypostasis as "substance" was in the book "On the cause of plants" by Aristotle's successor Theophrastus (c. 371 - c. 287 BC), while the term may have been introduced in the philosophical discourse either by the Stoic Poseidonius (c. 135 BC - c. 51 BC) according to some, or by the Epicurean Demetrius Lacon (fl. late 2nd century BC) according to others, in both cases with the meaning of objective or concrete existence or reality. Thus, real entities were said to "have hypostasis", whereas merely apparent or imaginary entities did not.
Recalling at this point that both the Stoic and the Epicurean philosophical schools were materialistic, the first conceiving the all-pervading Logos as a subtle fiery aether and the second conceiving reality as consisting just of atoms and void, it is clear that, to the extent that they used the term ousía, they would not have used it in the sense of form understood as universal or even in the sense of form at all, but in the sense of the individual, particular, concrete entity, the "hypokeimenon", “that which underlies or lies beneath”, which clearly overlaps with the meaning of hypostasis. Therefore, it is clear that for both Stoics and Epicureans the terms ousía and hypostasis were synonyms, but that was due to the specific sense in which both schools understood ousía.
Introducing now the Latins, who were now becoming interested in philosophy, into the picture:
Since, per linguistic design, substantia = hypostasis,
and, per Stoic and Epicurean understanding of ousía, hypostasis = ousía,
then, for the Latin-speaking Stoic- and Epicurean-dominated culture, substantia = ousía.
This was precisely the case of Tertullian, who was heavily influenced by Stoic metaphysics and coined the formula "tres Personae, una Substantia" early in the III century, after which substantia came to mean, among Latin-speaking Christians, the single infinite spirit that the Father, the Son and the Holy Spirit have in common, which is the meaning of ousía in the Nicene sense, as we will see next.
5. The "one ousía" answer to Arianism, its modalistic and tritheistic risks, and Nicaea's "one hypostasis" as prevention of the second.
Since Arians denied that the Son and the Father were one, they had to interpret Jesus' words "I and the Father are one." (Jn 10:30) in the sense of only moral oneness and not ontological oneness. To counter that eisegesis, the orthodox sought for the right word to add to Jn 10:30 to convey unequivocally the sense of ontological oneness: "I and the Father are one"... what?
We arrive at the same question from four key statements by Jesus of his divinity that were not mentioned in the previous section: the 4 times in which He explicitely applied to Himself the divine Name in the first person revealed in Ex 3:14: "Ehyeh", "Ego Eimí", I Am": Jn 8:24,28,58 & 13:19. If each of the Father and the Son names Himself "Ego Eimí", "I Am", then Each is a distinct "I" but Both are the same... what?
The natural answer to that question was the term ousía, which derived precisely from the verb "eimí", "to be". So, the initial orthodox answer to the Arian challenge was: the Father and the Son are one ousía and two prósōpa, one being and two persons. Now, each term in that answer, ousía and prósōpon, was at risk of misinterpretation resulting in another heresy.
The risk of misinterpretation of ousía came from the fact that the philosophical stage at the beginning of the IV century was very different from that of around 70 AD when the Letter to the Hebrews was written. Platonism had come back with Plotinus (204-270) and Porphyry (234-305), and with it the notions of forms and universals and therefore the issue of the exact meaning of ousía. Let us recall from section 1 that ousía had entered the philosophical discourse, some 700 years before Nicaea, not as a definition but as a question to be answered, and for which Aristotle had provided two different answers, one in the Categories and another in the Metaphysics, the second of which could in turn be interpreted in two different ways:
C: the individual, particular, concrete subject or "hypokeimenon", “that which underlies or lies beneath”;
M1: the essence or form, understood as particular;
M2: the essence or form, understood as universal.
Each of these senses of ousía led to a different sense of the statement that the Father and the Son were "homoousious", "of the same ousía":
C: They were the same particular, concrete subject.
M1: They had the same essence or form, understood as particular.
M2: They had the same essence or form, understood as universal.
It is easy to see that, while sense C amounts to modalism, sense M2 amounts to tritheism, since the case would be the same as stating that three horses have the same essence or form. It was probably to prevent the risk of understanding ousía as a universal of which there were multiple instances, or equivalently, of understanding the expression "of the same ousía" in the sense of only qualitative identity instead of numerical identity, that the Nicaea fathers added hypostasis - which, as we saw in section 2, had the meaning of objective or concrete existence or reality - as synonym of ousía, anathematizing those who asserted that the Son of God "is of another hypostasis or ousía"  . Since substantia was originally the cognate and equivalent of hypostasis, making hypostasis equivalent to ousía (M1) reinforced the Western understanding of substantia as the Latin equivalent of ousía (M1) that had begun with Tertullian.
Whereas the addition of hypostasis as synonym of ousía effectively prevented the M2 sense of ousía as universal, which leads to tritheism, it in effect promoted the C sense - which is evident from the etymological equivalence of hypostasis, literally "that which stands under", with hypokeimenon - which leads to modalism.
Which takes us precisely to the risk of misinterpretation of prósōpon, which came from the fact that it meant "face", "mask" or "character in a theatrical play" . Thus modalists could claim that they were in complete agreement with the definition of Nicaea.
6. The three Hypostases formula: appearance, toleration by St. Athanasius and resistance by St. Jerome.
As noted in section 3, the use of hypostasis as synonym of ousía, so that the Father and the Son were said to be "of the same hypostasis", was inconsistent with the sense of hypostasis in Heb 1:3, since that passage necessarily implies - from the very notion of "charaktēr" as impressed image or copy, reproduction, representation - that the Son is or has a numerically different hypostasis from that of the Father, irrespective of what hypostasis may mean.
It was not this consideration, however, what motivated orthodox theologians to advocate formulating trinitarian doctrine in terms of three hypostases, but the realization that using all the terms that can convey a sense of objective reality - i.e. hypostasis and ousía - to denote the common being of the Father and the Son amounted to leaving the door open for modalists to claim that they were in full agreement with the dogmatic definitions of the Church.
The notion of three Hypostases had been introduced in Christian theology by Origen and had been used also by his former student St. Dionysius of Alexandria. Its first documented use after Nicaea was by (also Alexandrian) St. Athanasius, in his work "In Illud Omnia", written probably c. 335  :
"For the fact of those venerable living creatures [Isa 6:3; Rev 4:8] offering their praises three times, saying 'Holy, Holy, Holy,' proves that the three Hypostases are perfect, just as in saying 'Lord,' they declare the one Ousía."
After that, the expression was not used again by St. Athanasius, in deference to the anathema of Nicaea , but was used by non-Nicene bishops, e.g. in the Dedication creed of the council of Antioch of 341 and in a letter by the homoiousian bishop George of Laodicea in 359. But by the time of the Council of Alexandria of 362, presided by St. Athanasius, the expression was already being used by people holding homoousian orthodoxy, as attested by the letter to the Church in Antioch written by that Council, known as "Tomus ad Antiochenos"  , which acknowledged that both expressions, "three hypostases" and "one hypostasis", could used in a sense consistent with homoousian orthodoxy.
In stark contrast with St. Athanasius in the Council of Alexandria of 362, St. Jerome, in his epistle 15 to Pope St. Damasus, written in 376 or 377   , manifests his deep trouble with the use of the formula "three hypostases".
7. The three Hypostases formula: proposal by St. Basil and St. Gregory of Nyssa and increasingly official Church adoption since 382.
After Nicaea, the first orthodox theologian to propose a notion of hypostasis distinct from that of ousía in a published work was St. Basil of Caesarea (330-379), and he does it in his epistles 214 (375) to Count Terentius  and 236 (376) to Amphilochius . In both letters, the main motive for using hypostasis as synonym of Person is that, if the orthodox keep speaking of one hypostasis, they set the stage for the Arians to accuse them of Sabellianism (modalism).
St. Gregory of Nyssa (335-394) devotes his epistle 35 (c. 380) addressed to his brother Peter - which is often referred to as St. Basil's epistle 38 to his brother Gregory - to the difference between ousía and hypostasis. St. Gregory starts by distinguishing between the common or general and the particular, and provides a definition of hypostasis .
In 381, i.e. around one year after Gregory's letter to Peter, the second Ecumenical Council convened in Constantinople, and the next year, i.e 382, a synod of bishops assembled in that city and sent a Synodical Letter to Pope Damasus and other Western bishops, which is the first official (though not ecumenical) Church document speaking of three Hypostases :
Letter from the Constantinople Synod of 382 to Pope Damasus
What we have undergone — persecutions, afflictions, imperial threats, cruelty from officials, and whatever other trial at the hands of heretics — we have put up with for the sake of the gospel faith established by the 318 fathers at Nicaea in Bithynia. You, we and all who are not bent on subverting the word of the true faith should give this creed our approval. It is the most ancient and is consistent with our baptism. It tells us how to believe in the name of the Father and of the Son and of the Holy Spirit: believing also, of course, that the Father, the Son and the Holy Spirit have a single Godhead and power and substance, a dignity deserving the same honour and a co-eternal sovereignty, in three most perfect Hypostases, or three perfect Persons. So there is no place for Sabellius’s diseased theory in which the hypostases are confused and thus their proper characteristics destroyed.
End of quote from the Constantinople Synod of 382.
In 431 the third Ecumenical Council convened in Ephesus. The third letter of St. Cyril of Alexandria to Nestorius, which was read at the council and included in the proceedings, spoke of the Hypostases of the Word and of the Holy Spirit :
St. Cyril of Alexandria, in his 3rd letter to Nestorius
All the expressions, therefore, that occur in the gospels are to be referred to one Person, the one enfleshed Hypostasis of the Word.
For even though the Spirit exists in his own Hypostasis and is thought of on his own, as being Spirit and not as Son, even so He is not alien to the Son.
End of quote from St. Cyril.
In 451 the fourth Ecumenical Council convened in Chalcedon and proclaimed a Christological definition which contains all the terms considered so far: ousía, prosopon, hypostasis, and nature  .
Finally, in 553 the fifth Ecumenical Council convened in Constantinople and proclaimed a series of trinitarian and Christological definitions, the first of which explicitely identifies ousía with physis and hypostasis with prosopon .
 Studtmann, Paul, "Aristotle's Categories", The Stanford Encyclopedia of Philosophy (Fall 2017 Edition), Edward N. Zalta (ed.)
 Cohen, S. Marc, "Aristotle's Metaphysics", The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.)
 Loux, Michael J., "Primary Ousia: an essay on Aristotle's Metaphysics Z and H", Cornell University Press, 2008.
 Cohen, S. Marc, "Z.13: Substances and Universals", 2008.
 Ute Possekel, "Evidence of Greek philosophical concepts in the writings of Ephrem the Syrian", Peeters Publishers, Louvain, 1999.
 Thomas G. Weinandy, Athanasius: A Theological Introduction, Ashgate Publishing, 2007.
 Lewis Ayres, Nicaea and Its Legacy: An Approach to Fourth-Century Trinitarian Theology, Oxford University Press, Oxford, Oct 28, 2004.
 The Letters of Saint Jerome, Aeterna Press, 2016.
For the quoted passage, I used this Latin text to improve the accuracy of the English translation in the previous two references.
 Richard Price and Michael Gaddis, The acts of the Council of Chalcedon, Liverpool University Press, 2005.
 Pavouris, Raphael (2001), The condemnation of the Christology of the three chapters in its historical and doctrinal context: the assessment and judgement of Emperor Justinian and the Fifth Ecumenical Council (553). PhD thesis, University of Glasgow.
Article from which the above has been extracted: