Recently finished Ravenna: Capital of Empire, Crucible of Europe and don't quite undestand the arguments they were having in late antiquity on the subject.
The period we call late antiquity was characterised, from a Christian perspective, by Christological debates. This term is about the nature of Jesus. Between 325 and 787 there were seven church councils that met to discuss various points of church doctrine, and questions on the nature of Christ were paramount amongst these conversations. Arianism was mainly discussed in the first two councils (Nicaea and Constantinople), and could be summed up as Jesus’ divine relationship as the son of God to God the Father, while later councils moved onto other Christological questions such as the relationship between Christ’s divine and human aspects.
The arguments themselves are very technical and hinge on seemingly minor details but theoretically have significant consequences. If you go back to the early church, including the New Testament documents, you can see that what Christians believe about Jesus is a bit fuzzy. They clearly hold him in very high regard – at times his followers proclaim him divine (e.g., Thomas’ exclamation to Jesus in John 20:28, “My LORD and my God!”), but at the same time he is human (on account of him having been born, growing up, eating, sleeping, weeping, and dying). Even Jesus himself makes statements that seem contradictory. John’s Gospel has what is called a high Christology, meaning that the divine aspects of Jesus are stressed. In John 10:30 Jesus claims that “I and the Father are one”, and the opening to the Gospel identifies Jesus as the ‘Word’ (Greek logos), and that “In the beginning was the Word, and the Word was with God, and the Word was God” (my emphasis). Within the first hundred years of Jesus’ death, which is typically when we think the New Testament documents were written, Jesus has two natures: human and divine.
What happens over the next few hundred years is that the church starts to expand rapidly, crossing vast cultural and geographical distances. Out of this context there are different versions of Christian belief that emerge and interact with each other. Very early on, within the first century and arguably present in some of the New Testament texts, certain figures in the church are very concerned that people should have the right sort of beliefs, and they take aim at those who think differently. The word heresy that we use today comes from Greek hairesis and it is used about nine times in the New Testament. Broadly speaking it comes from a word meaning ‘choice’ and by the first century and it is used about nine times in the New Testament. Broadly speaking it comes from a word meaning ‘choice’ and by the first century AD meant 'sect' or 'party', usually in a negative sense, implying a group who had made the 'wrong choice'.
There is ongoing debate about the state of Christianity in the centuries between the first and the fourth, but most scholars see it as a time of establishing parameters on a range of issues, including leadership structure and beliefs. Bart Ehrman argues that the group which becomes the orthodox church in the fourth century is present as ‘proto-orthodoxy’ prior to Nicaea but that they are simply one of a number of competing, but equally valid versions of Christianity (hence the title of his 2003 book Lost Christianities). Other academics push back on this and believe the ‘proto-orthodox’ branch to have legitimate claim to originality over the other groups which are divergent (e.g., Arland Hultgren).
This debate is of central importance to your original question as it determines how you understand the terms heresy and orthodoxy. Orthodoxy means ‘correct belief’ and while it is possible to analyse it as an historical development, there is a theological element to it as well. As an historical phenomenon, I suggest that orthodoxy is the side that emerges with a majority consensus but this is a relativistic position, in part because the dominant view changes at different points in time. Probably the game changer is Constantine coming to power and legalising Christianity with the Edict of Milan in 313. Prior to this Christianity had developed through individual figures in the church arguing with each other and with those they see as heretics but there is no centralised authority to regulate this. When Constantine adopts Christianity and calls the first council, Nicaea in 325, things change.
Arianism, accredited to the presbyter Arius of Alexandria in Egypt, makes a lot of sense at face-value. As already shown in the New Testament texts, early Christians have a range of beliefs on the humanity and divinity of Christ. The Arian controversy focuses on the divine aspect by raising questions about what it means for Jesus to be divine and for Christianity to still be a monotheistic religion. I mentioned the high Christology of John’s Gospel, but there are statements by Jesus in the other Gospels that imply his subordinate role to God. In Mark 10:18 he replies to a rich man who calls him ‘good’ with “Why do you call me good? No one is good but God alone”. When asked by his disciples about the end of the world, Jesus claims that “about that day and hour no one knows, neither the angels of heaven, nor the Son, but only the Father” (also in Mark 13:32). These kinds of passages led Arius to argue that, while still divine, Christ was in some sense subordinate to and created by God the Father.
Arius’ views gained some popularity. Certain Gothic tribes adopted it and even some of Constantine’s children, however at the time of the council it did not garner widespread support. We do not know that much about what individual bishops at the council thought, not even the exact number of attendees is known, but by one account at least, of the 300 or so bishops there were only two who did not agree to condemn Arius’ views as heretical. The council drafted the Nicene Creed and defined the orthodox position. Christ is “true God from true God, begotten not made, of one essence with the father”. The term for the latter statement from the Nicene Creed here is homoousia, a word that means ‘of the same essence’, or ‘of the same being’ (homo meaning ‘the same’ and ousia meaning ‘being, essence, nature’).
To travel back to the original question, Arianism was heretical then, in the sense that it was a minority position. This works if orthodoxy is defined by a majority consensus that wields some form of power, which is what emerges post-Nicaea. However, it is not the case that the Catholics won the argument and wrote history to favour themselves. For starters, the victory was a long time coming. The first history of the church proper was written by Eusebius of Caesarea (published around the 330s in as Historia Ecclesiae), and it seems that he had Arian leanings. Further, after Constantine’s death, the balance of power swung back and forth, with Constantius II, one of Constantine’s sons, supporting Arian bishops over Nicene bishops like Athanasius. Nicene orthodoxy does win out in the end, but it mostly due to the work of those who supported it rather than them writing Arianism out of history. And even though there were supporters of Arianism, and powerful ones at that, it was always a minority position.