In fact, maybe it could be helpful to offer some sort of explanation and history behind some of the language here.
Perhaps the key word here is hypostasis (plural hypostases). It's a fairly mundane Greek word, in some contexts it could be used to describe the sediment that fell to the bottom of a container of standing liquid. It is literally the conjoining of two other Greek words: hypo meaning "under", and "stasis" meaning "standing [still]", as such a sediment at the bottom of a liquid is hypostasis, the stuff under the liquid in a container of standing, still liquid. But the word also had some use in philosophical circles, where it was used sometimes as meaning a particular thing, a substance or subsistance. In philosophical circles it was sometimes used interchangeably with the word ousia, meaning "being".
In the early centuries of Christianity as various Christological debates broiled and bubbled, and it became increasingly necessary to use more precise, well-defined language, these words hypostasis and ousia gained a particular use in theology.
Sabellianism pushed things in this way considerably, since for Sabellius, Praxeas, and Noetus the word "hypostasis" and "ousia" were basically interchangeable. So they said that God was a singular hypostasis of one ousia, what we behold merely by perception is God revealing Himself through three prosopa ("masks" or "faces"). This introduces another word, and it's another important one: prosopon (plural prosopa) which, as I said means "mask" or "face", it's standard Latin equivelant, however, is persona (plural personae) which is where English gets "person".
The word hypostasis was used to describe the specific, particular, distinct "that" we were referring to when speaking of the Father, Son, or Holy Spirit. That is, to speak of the hypostasis of the Father, or the hypostasis of the Son, and that these were two distinct hypostases. That is, the Father is a distinct that from the distinct that of the Son. Because Father-ness and Son-ness aren't the same, there is a that from which the other that is distinct. The Father begat the Son, but the Son did not begat the Father, since begetting is unique to that which the Father is, and begotten is unique to that which the Son is.
So the Father begat the Son.
The Son is begotten of the Father.
Now the 4th century heresiarch Arius taught that this, that the Father has begat the Son, that the Son is begotten of the Father, meant also that the Son must therefore be less than the Father. The Son must be a creature. Now, the Son may be the most powerful and profound creature, greater than all other creatures by infinite magnitudes--but still a creature. In fact the Son is so glorious that Arius and the Arians still called Him God, it's just that the Son was God in a very different way than the Father was. Since the Son wasn't eternal, but created.
Arius wasn't saying this just to be a contrarian. Arius was a student of a very famous Christian teacher from Antioch, Lucian of Antioch. Lucian was famous for his opposition against Sabellianism. As such, Arius was, in his own way, trying to insure that Sabellianism wasn't cropping its head up over there in Alexandria where he was presbyter.
For Arius, therefore, the Father and the Son were very much distinct hypostases, and their ousia was perhaps similar but ultimately different. As such the Arians liked to say that the Son is heteroousia (of a different being than the Father), or perhaps softening it up more by saying the Son is homoiousia (of a similar being as the Father).
When these debates started to really get heated, according to a contemporary historian one couldn't even go to the market to buy bread without getting into an argument with someone about whether the Son was of the same being or a different or similar being. These things largely only were going on in the eastern half of the Roman Empire however, as Arius had been in Egypt and then moved to Palestine. The Roman emperor, Constantine, who had made the Christian religion legal, and recently just unified both halves of the Roman empire under his rule, and was giving Christianity a large deal of imperial patronage saw these debates as problematic. So he figured that if he just made all the bishops come together at one spot and debate it at one big council that they should be able to figure it out. Constantine had no care one way or the other about the direction the council would go, he had no dog in that fight, he just wanted a resolution regardless of what the resolution was.
And so, in the summer of 325 AD about 300 bishops convened at the city of Nicea near Constantinople. And over the course of the length of the council, the bishops argued, and argued, and argued, and argued. But as things were going, something of a consensus was emerging among many, that neither a Sabellian nor Arian position could be acceptable. And a particular word was being thrown around, though it was controversial (because some Sabellians had also used it), that word was homoousia (of the same being). And, eventually, most of the bishops were able to agree to a written symbol of faith--a creed.
Of course the council didn't actually settle the debate, in fact some of Constantine's closest spiritual advisors and friends were themselves Arian or sympathetic to Arianism/Arius, including his biographer Eusebius of Caesarea, and the man who would eventually baptize Constantine on his deathbed, Eusebius of Nicomedia. After Nicea Constantine would come to largely embrace and support the Arian cause, exile pro-Nicene bishops like Athanasius, and after his death the two sons of Constantine would be themselves split over this very issue, though the Arian son would ultimately gain power, and Arianism would have a lot of political power for most of the 4th century until Julian the Apostate.
In all of this, a particular language emerged to speak of God: That there are three Hypostases of one Ousia, and therefore the three are, indeed, one. The Father is God, the Son is God, and the Holy Spirit is God, not by a confusion, but by their unity of Being. For what the Father is, the Son is also; and what the Father and the Son are, the Spirit is also. For if the Father is God, then the Son is also God, not another god, but one and the same God. If the Being is one, then the Son can be nothing other than what the Father is, God of God, truly God of truly God. Indeed:
"We believe ... in one Lord Jesus Christ, the Son of God, begotten of the Father, the only-begotten; that is, of the being of the Father, God of God, Light of Light, true God of true God, begotten, not made, of the same being as the Father; by whom all things were made both in heaven and on earth." - from the text of the 325 Nicene symbol
-CryptoLutheran
