The Catechism of the Catholic Church defines these three sins against the faith in this way:
2089 Incredulity is the neglect of revealed truth or the willful refusal to assent to it.
"Heresy is the obstinate post-baptismal denial of some truth which must be believed with divine and catholic faith, or it is likewise an obstinate doubt concerning the same;
apostasy is the total repudiation of the Christian faith;
schism is the refusal of submission to the Roman Pontiff or of communion with the members of the Church subject to him." [Code of Canon Law c.751]
The Church's moral theology has always distinguished between objective or material sin and formal sin. The person who holds something contrary to the Catholic faith is materially a heretic. They possess the matter of heresy, theological error. Thus, prior to the Second Vatican Council it was quite common to speak of non-Catholic Christians as heretics, since many of their doctrines are objectively contrary to Catholic teaching. This theological distinction remains true, though in keeping with the pastoral charity of the Council today we use the term heretic only to describe those who willingly embrace what they know to be contrary to revealed truth. Such persons are formally (in their conscience before God) guilty of heresy. Thus, the person who is objectively in heresy is not formally guilty of heresy if 1) their ignorance of the truth is due to their upbringing in a particular religious tradition (to which they may even be scrupulously faithful), and 2) they are not morally responsible for their ignorance of the truth. This is the principle of invincible ignorance, which Catholic theology has always recognized as excusing before God.
The same is true of apostasy. The person who leaves not just the Catholic Church but who abandons Christ Himself is materially an apostate. He is formally an apostate through willful, and therefore culpable, repudiation of the Christian faith.
Finally, the person who refuses submission to the Roman Pontiff, whom Vatican I defined as having a universal primacy of authority over the whole Church, is at least a material schismatic. It was thus common in the past to speak of the schismatic Orthodox Churches who broke with Rome in 1054. As with heresy, we no longer assume the moral culpability of those who belong to Churches in schism from Rome, and thus no long refer to them as schismatics.
When it comes to Catholics who are formally guilty of heresy, apostasy or schism, the Church applies the penalty of excommunication. The 1983 Code of Canon Law, repeating the sanctions of the earlier 1917 Code, states,
1. With due regard for can. 194, part 1, n. 2, an apostate from the faith, a heretic or a schismatic incurs automatic (latae sententiae) excommunication and if a cleric, he can also be punished by the penalties mentioned in can. 1336, part 1, nn. 1, 2, and 3.
2. If long lasting contumacy or the seriousness of scandal warrants it, other penalties can be added including dismissal from the clerical state.
This canon is saying that once a person willingly repudiates Christ, embraces a heresy, knowing it to be contrary to divine and Catholic faith, or refuses submission to the Roman Pontiff (or communion with the members of the Church subject to him), by virtue of the law itself they are automatically excommunicated. No ecclesiastical act is necessary and no public notice.
However, to incur this latae sententia excommunication one must satisfy the general conditions for canonical culpability set out in the Code. For example, a person who has not been diligent (prudently weighing the issues involved) in their action is not punished.
1. No one is punished unless the external violation of a law or a precept committed by the person is seriously imputable to that person by reason of malice or culpability.
2. A person who has deliberately violated a law or a precept is bound by the penalty stated in the law or that precept; unless a law or a precept provides otherwise, a person who has violated that law or that precept through a lack of necessary diligence is not punished.
3. Unless it is otherwise evident, imputability is presumed whenever an external violation has occurred.
A person who lacks the proper use of reason is likewise not punishable.
c. 1322 Persons who habitually lack the use of reason are considered incapable of an offense even if they have violated a law or a precept while appearing to be sane.
The following canon completes the list of conditions that can prevent the application of an excommunication and other ecclesiastical sanctions.
The following are not subject to penalties when they have violated a law or precept:
(1) a person who has not yet completed the sixteenth year of age;
(2) a person who without any fault was unaware of violating a law or precept; however, inadvertence and error are equivalent to ignorance;
(3) a person who acted out of physical force or in virtue of a mere accident which could neither be foreseen nor prevented when foreseen;
(4) a person who acted out of grave fear, even if only relatively grave, or out of necessity or out of serious inconvenience unless the act is intrinsically evil or verges on harm to souls;
(5) a person who for the sake of legitimate self-defense or defense of another acted against an unjust aggressor with due moderation;
(6) a person who lacked the use of reason with due regard for the prescriptions of cann. 1324, part 1, n. 2 and 1325;
(7) a person who without any fault felt that the circumstances in nn. 4 or 5 were verified.
The penalty of excommunication carries with it the prohibition of receiving the sacraments, while not excusing from obligations such as Sunday and Holy Day Mass, Easter Duty etc... To be reconciled to the Church a person who has been excommunicated, even if that fact is known to the person alone, must first have the excommunication lifted. With the exception of certain crimes reserved to the Holy See, each bishop has the authority to remit the penalty of excommunication. However, he generally delegates this faculty to his priests, or at least to certain confessors (usually at the Cathedral).
Thus, by going to confession one can usually have the penalty lifted. If recourse to higher authority is needed the confessor will say so and invite the person to return the following week or at another time. He will then obtain the remission of the penalty from the bishop (protecting the person's anonymity, of course.) and communicate it to the penitent. The person is then free to make a good confession and be fully reconciled with Christ and the Church.
The Special Danger of Ultra-Traditionalist Movements
There is within the Church today a special danger for those who, often for seemingly legitimate reasons (abuses of the liturgy, the open promotion of heresy even by clergy, and similar causes), have sought refuge in traditionalist movements on the margins of the Church. These groups, distinguishable from those who love the Tridentine tradition of the Mass and sacraments and who celebrate them in Communion with the Pope, go their own way outside of the laws of the Church. They typically rationalize their disobedience by attacking the Second Vatican Council, the current liturgical rites, ecumenical and interreligious dialogue, and often Pope John Paul II personally, never distinguishing between teaching and law on the one hand, and the abuse of it by dissenters and the disobedient on the other.
These groups, such as the Society of St. Pius X, of Pius V, the "We Resist You to Your Face" movement, Br. Dimond and Holy Family Monastery, make ready use of scandals to gain support among the unwary, who, discouraged by their local situation, may think they are joining a more perfect orthodoxy and a more loyal remnant of Catholics. Thankfully such motives may excuse the average person who takes comfort in such groups, at least initially, though as St. Thomas Aquinas teaches to take scandal in other's sins is istself sinful. However, there is a great danger that starting from the material schism of refusing submission to the Pope, that all these groups have in common, the Catholic cannot long maintain the schizophrenic position of saying they are being submissive to the Pope while disobeying him. At some point they must choose and formally adhere to the schism of the group. In some cases the group identity depends on some formal repudiation of the "Novus Ordo" Church, very effectively hastening the spiritual demise of the lay adherent.
Also unfortunate for such souls is the fact that these ultra-traditionalist groups profess to be doctrinally orthodox, an orthodoxy which necessarily includes the teaching that Outside the Church There Is No Salvation. This means that someone who has formally separated himself from the Church through heresy or schism, or knowing the Church to be true failed to enter her, cannot be saved, unless of course they renounce their own will and reconcile with the Church. Unlike the non-Catholic Christian, can the super-orthodox claim invincible ignorance of this teaching? Can they escape the condemnation of Pope Boniface VIII, who in first elaborating it said, "this authority, although it is given to man and is exercised by man, is not human, but rather divine, and has been given by the divine Word to Peter himself and to his successors in him, whom the Lord acknowledged an established rock, when he said to Peter himself: Whatsoever you shall bind etc. [Matt. 16:19]. Therefore, whosoever resists this power so ordained by God, resists the order of God ...? No wonder that given enough time such groups inevitably produce those who claim that the See of Peter is vacant, since the logic of their schismatic attitude is ultimately irreconcilable with the doctrine of papal primacy, as enunciated by both Pope Boniface and Vatican I.