When we talk about the Trinity, we have to be careful to use precision in our language. This is because when we do not, things get more confusing than they already tend to be. And false teachers have historically exploited this confusion to propagate heresy and lead people astray.
Perhaps for this more than any other reason, the ancient church ended up coming up with a set of well-defined terms to use in order to speak precisely about the Trinity. When we learn and use those terms, it allows us to speak about the Trinity in a way that is less likely to be misunderstood, and denies heretics the possibility of finding 'wiggle-room' in semantics (ambiguity allows them to say that they agree with you because they are using the same words to mean something different than you do).
So lets review a few terms that through church history have developed a very well-defined meaning:
Person/Hypostasis: "Hypostasis" is a transliteration of the Greek word historically equated with the English word "person". On the Orthodox Church of America's website they give a helpful explanation of the term: "In Orthodox terminology the Father, the Son and the Holy Spirit are called three divine persons. Person is defined here simply as the subject of existence and life—hypostasis in the traditional church language."
Substance/Ousia: Merriam-Webster defines "ousia" as "true being: entity, essence, substance". Historically the terms "nature", "being", "Godhood", and "Godhead" all end up being used equivalently along with "essence" and "substance" to describe the divine nature of the three persons of God the Father, Christ, and the Holy Spirit.
Heteroousios: different substance; from the prefix "hetero" meaning "different" and "ousia" meaning "substance". Used by the Arians to describe their Christ as not being of the same divine nature as the Father as thus not God.
Homoousios: same substance; from the prefix "homo" meaning "same" and "ousia" meaning "substance". Used by the Orthodox to describe Christ as being of the same divine nature as the Father, and thus truly God. Often translated into English as "co-essential" or "consubstantial".
Consubstantial: of the same substance. (Equivalent "co-essential"). Used along with homoousias to describe that something is of the same nature/essence/substance, ei the Orthodox formula that 'Christ is consubstantial with the Father' means He is of the same substance/divine nature as the Father. According to the ancient Chalcedonian Definition Christ is also consubstantial with mankind: "co-essential with the Father according to the Godhead, the Self-same co-essential with us according to the Manhood". This was used to describe Christ's true divinity and humanity.
Godhood/Godhead: divinity. In orthodox discussion of the Trinity it often ends up being used the same way as "substance" and "essence" to refer to the divinity/divine nature of the persons of the Trinity.
Now that we have the vocabulary to accurately speak about the biblical realities of the Trinity, lets examine Arianism, Sabellianism, and Orthodoxy.
One way we can break down these three views of the Trinity is in terms of persons and nature:
Orthodoxy says that there is one Godhood existing in three distinct persons -Father, Son, and Holy Spirit. In this understanding, God the Father and His only-begotten Son are numerically distinct persons (that is, the Father is not the Son and the Son is not the Father) and are of the same essence/divine nature (so the Son is God in nature just as the Father is). So in terms of person they are distinct, but in terms of nature they are the same (homoousios).
Sabellianism (also known as Modalism) is an ancient heresy that agrees with Orthodoxy that Christ is of the same nature as the Father -but adds to that that they are not only the same in nature, but that they are actually the same person. It teaches that there is only one divine person who acts at various times in different personas or modes -sometimes revealing himself as Father, other times as Son, other times as the Spirit. But at the end of the day, it says, they are all one and the self-same person simply acting in different roles.
Arianism is another ancient heresy that veers off the Orthodox path in the opposite direction from Sabellianism. It agrees with Orthodoxy that Christ and the Father are distinct persons -but goes beyond Orthodoxy by extending the difference to their natures as well, asserting that the Father and Son are distinct in person and in nature/essence (heteroousios).
So in summary, Orthodoxy says the Father and Son are distinct persons of the same nature (consubstantial); Sabellianism admits that the Father and Son are of the same nature but denies the distinction in persons, while Arianism admits the distinction between the persons of the Father and the Son but denies their consubstantiality. In this respect, Sabellianism and Arianism are sort of mirror heresies, each taking the opposite extreme, with Orthodoxy in the middle.
So why study these heresies? Why even bother remembering what they are? Aren't they ancient problems that the ancient church dealt with and put down, yesterday's news? How does this have any bearing whatsoever on the average Christian today?
Although the heresies of Arianism and Sabellianism were defeated by the early church and eventually died out completely or near-completely in the centuries following the Nicene Council, like theological zombies, they eventually returned from the dead to trouble the church afresh following the Protestant Reformation. Today many manifestations and variations of these heresies can be found again assailing the church, and we need to know what they are so we can be on alert against them. Arianism has risen from the grave to devour souls again in the modern 'Jehovah's Witness' cult, and Sabellianism again plagues the church in such forms as Pentecostal 'Oneness Theology'.
Both of these ancient heresies are still active dangers to the church today, and one of the greatest advantages they have is the ignorance of the average professing Christian. To quote one of my favorite movies (The Usual Suspects) "The greatest trick the devil ever pulled was convincing the world he didn't exist". People are most susceptible to the dangers they never realize are a threat. Ignorance of heresy opens the door for that heresy. But far more dangerous than ignorance of heresy is ignorance of Orthodoxy. Know Orthodoxy, and heresy will be unable to deceive you. But remain ignorant of it, and you may soon find yourself confessing something different than what the Bible teaches and the church has confessed.
So like with zombies, you don't need to study each individually in order to find a new way to kill them, you just need a good shotgun and some ammo; so also with trinitarian heresies we need only take up the shotgun of the Nicene Creed and the ammunition of Nicene Orthodoxy to put down these doctrinal aberrations. Of course this is no replacement for the sword of the Spirit, the word of God, but rather let us wield both these weapons, like what Paul wrote, wielding "weapons of righteousness for the right hand and the left" against the myriad variations of these heresies that assail the church today.
So in conclusion, and if you take nothing else away from this, studying Nicene Orthodoxy matters and has great practical value to Christians because it protects us from the various heresies in the world that are constantly assailing the church in an attempt to lead whoever they may astray from the truth. And if we love God it is only natural that we should seek to understand Him as He has revealed Himself to us, that we might worship Him, serve Him, and glorify Him in truth and in knowledge, for the glory of God shines brightly in the truth of who He is.
May our love for God our Father and Christ our Lord spur us to strive to better and more accurately understand Them, and may our understanding of the truth of who They are in turn fuel our love for Them all the more.
"...contend earnestly for the faith which was once for all handed down to the saints." (Jude 1:3 NASB)
Grace and peace,