There are two types of heresy: material and formal.
Material heresy is any idea (assertion, teaching, claim, etc.) that contradicts, or is fundamentally incompatible with, any magisterial teaching that must be believed with the full assent of faith (divine and catholic faith). Every infallible teaching of the Magisterium must be believed with this full assent (also called theological assent).
Formal heresy is the knowing choice to believe (or even teach) material heresy. Everyone who commits formal heresy is automatically excommunicated by the very nature of the offense; the act of formal heresy cuts the heretic off from the Church. Another way of describing this same sin is to say that the formal heretic obstinately doubts or obstinately denies a teaching that must be believed with divine and catholic faith (full assent).
However, a person might believe or teach material heresy, mistakenly thinking that the idea is either a correct understanding of magisterial teaching or a sound theological opinion that is compatible with magisterial teaching.
A person who believes or teaches mere material heresy does not realize that he is believing or teaching heresy. A person who commits the sin of formal heresy does realize, in one way or another, that he is rejecting a definitive teaching of the Magisterium. He might not label his act ‘heresy’. He might feel justified in thinking that he is right and the Church is wrong. But in order to commit the sin of formal heresy, the person must realize that the teaching being obstinately doubted or obstinately denied is a definitive teaching of the Magisterium.
Many Catholics today, especially online, are teaching material heresy. In some cases, it is clear they realize that their own teachings are contrary to definitive teachings of the Magisterium. If the individual makes this clear, that he is knowingly choosing to reject an irreformable teaching of the Magisterium, then he is making it clear that he has committed formal heresy.
However, in many other cases, it seems that the individual has simply badly misunderstood the infallible teaching of the Magisterium. There are very many Catholics who are using the internet to teach material heresy as well as various doctrinal errors that fall short of heresy. These individuals often portray their misunderstandings of magisterial teaching as if it were a correct understanding. The result is much confusion among the faithful; they do not know whom they should believe. This problem is particularly prevalent in online Catholic discussion groups and in online blogs.
I was formerly a member of one discussion group, in which the moderators (most of whom were anonymous) were each presenting themselves as teachers of the Catholic Faith. They taught many doctrinal errors. At one point, moderator ‘A’ was telling me that the Church teaches that the Bible contains errors, except on matters pertaining to salvation; and that this is a required belief. I would be kicked out of the group if I disagreed. And at the same time, in a different section of the same discussion group, moderator ‘B’ was telling me that the Church teaches that the Bible contains no errors on any subject; and that this is a required belief. I would be kicked out of the group if I disagreed. So I resigned from the group.
By the way, moderator ‘B’ was correct. The Magisterium infallibly teaches that the Bible is totally inspired and totally inerrant. Every type of error on any subject is incompatible with inspiration. Since God is Truth, He cannot inspire error or falsehood.
Another problem with the teachings put forward by moderators in that same discussion group, was the claim that magisterial teachings can never err; they are supposedly inerrant. But the Magisterium itself teaches that only its infallible teachings are certain to be without error. The Magisterium teaches infallibly in any of three ways:
1. Papal Infallibility (solemn definitions of the Roman Pontiff)
2. Conciliar Infallibility (solemn definitions of Ecumenical Councils)
3. the ordinary and universal Magisterium
All other magisterial teachings, whether of a Pope or a Council or any Bishop or group of Bishops, are non-infallible and non-irreformable. Non-infallible teachings are subject to a limited possibility of error and of reform.
There exists in the Church a freedom of search for truth as well as norms for licit theological dissent. All persons are bound to seek the truth on matters of faith and morals. And, by virtue of divine law, they have the moral obligation and the moral right to embrace and observe the truth which they have come to know.
A faithful theologian might responsibly dissent from a non-infallible teaching, if he has a serious and well-founded basis in the other teachings of the Magisterium and in the teachings of Sacred Tradition and Sacred Scripture. A non-infallible doctrine may admit of development and may call for clarification or revision.
Only infallible teachings require the full assent of faith. Non-infallible teachings require a different type and degree of assent called the religious submission of will and intellect. Dissent from a non-infallible teaching can be licit and responsible.
Yet another problem in the same discussion group was that they taught a false doctrine: that we are only to learn from the Magisterium, that we should not learn directly from Sacred Tradition and Sacred Scripture, since we might err in our understanding. They promoted the view that any theological argument based on Sacred Tradition or Sacred Scripture should be disregarded, because the individual might err, whereas (they claimed) the Magisterium is always inerrant.
I departed from that group because anonymous moderators were teaching these and many other doctrinal errors to the group members.