St. Thomas Aquinas explains "conscience" (conscientia) in Summa Theologica I q. 79 a. 13 c.:

Properly speaking, conscience is not a power, but an act. This is evident both from the very name and from those things which in the common way of speaking are attributed to conscience. For conscience, according to the very nature of the word, implies the relation of knowledge to something: for conscience may be resolved into "cum alio scientia," i.e. knowledge applied to an individual case. But the application of knowledge to something is done by some act. Wherefore from this explanation of the name it is clear that conscience is an act.

The same is manifest from those things which are attributed to conscience. For conscience is said to witness, to bind, or incite, and also to accuse, torment, or rebuke. And all these follow the application of knowledge or science to what we do: which application is made in three ways. One way in so far as we recognize that we have done or not done something; "Thy conscience knoweth that thou hast often spoken evil of others" (Eccles. 7:23), and according to this, conscience is said to witness. In another way, so far as through the conscience we judge that something should be done or not done; and in this sense, conscience is said to incite or to bind. In the third way, so far as by conscience we judge that something done is well done or ill done, and in this sense conscience is said to excuse, accuse, or torment. Now, it is clear that all these things follow the actual application of knowledge to what we do. Wherefore, properly speaking, conscience denominates an act. But since habit is a principle of act, sometimes the name conscience is given to the first natural habit—namely, 'synderesis': thus Jerome calls 'synderesis' conscience (Gloss. Ezech. 1:6); Basil [*Hom. in princ. Proverb.], the "natural power of judgment," and Damascene [*De Fide Orth. iv. 22] says that it is the "law of our intellect." For it is customary for causes and effects to be called after one another.

^{("Synderesis"—from St. Jerome's "scintilla conscientiæ," a "prick of conscience"—is the part of conscience that guides future actions, and "syneidesis" is that part that evaluates past actions.)}

Thus, conscience is the act of applying abstract principles (e.g., "Thou shalt not commit adultery") to particular situations in one's life.

What that article seems to be implying that Francis describes is not conscience but the rationalization of an act, which is to come up with a general principle from one's actions, not to apply a general principle to one's actions. Conscience is to deduction as rationalization is to induction.

For an extended treatment on morals and Conscience, the current Catechism does not lack for material¹. Regarding your first question (the second is worthy of its own questoin):

1. Sin is a matter of violating God's law.

CCC 1849 Sin is an offense against reason, truth, and right conscience; it is failure in genuine love for God and neighbor caused by a perverse attachment to certain goods. It wounds the nature of man and injures human solidarity. It has been defined as "an utterance, a deed, or a desire contrary to the eternal law."
CCC 1850 Sin is an offense against God: "Against you, you alone, have I sinned, and done that which is evil in your sight." Sin sets itself against God's love for us and turns our hearts away from it. Like the first sin, it is disobedience, a revolt against God through the will to become "like gods," knowing and determining good and evil. Sin is thus "love of oneself even to contempt of God." In this proud self-exaltation, sin is diametrically opposed to the obedience of Jesus, which achieves our salvation.

2. Where does conscience fit in?
   I. The Judgment of Conscience

   MORAL CONSCIENCE 1776 "Deep within his conscience man discovers a law which he has not laid upon himself but which he must obey. Its voice, ever calling him to love and to do what is good and to avoid evil, sounds in his heart at the right moment.... For man has in his heart a law inscribed by God.... His conscience is man's most secret core and his sanctuary. There he is alone with God whose voice echoes in his depths."//snip//
   1782 Man has the right to act in conscience and in freedom so as personally to make moral decisions. "He must not be forced to act contrary to his conscience. Nor must he be prevented from acting according to his conscience, especially in religious matters."

   The role of Conscience is to speak to us, deep in our heart of hearts, and help us discern when we are confronted with sin, and help us make the right choice: to avoid sin and the near occasion of sin.

¹ PART THREE: LIFE IN CHRIST // SECTION ONE: MAN'S VOCATION LIFE IN THE SPIRIT // CHAPTER ONE THE DIGNITY OF THE HUMAN PERSON // ◾Article 4 THE MORALITY OF HUMAN ACTS. It cites a variety of theology, to include Thomas Aquinas (a Doctor of the Church).

In order to answer this question, it is necessary to understand precisely what is meant by “conscience” and its relationship to human acts (that is, those actions that can be qualified as morally right or wrong).

The Church, generally taking its cue from Medieval Scholasticism (see, e.g., Summa theologiae [S.Th.], Ia, q. 79, a. 13), defines the conscience in these terms:

Conscience is a judgment of reason whereby the human person recognizes the moral quality of a concrete act that he is going to perform, is in the process of performing, or has already completed. (Catechism of the Catholic Church [CCC] 1778; see also no. 1796. The number goes on to describe some of the properties of the conscience, but this is the actual definition. For a more detailed overview, see Veritatis Splendor [VS] 57-61.)

Some clarification of terminology is in order. The human intellect (according to the most common model used in Catholic philosophy) is fundamentally capable of two kinds of acts: apprehension, by which a person knows what sort of thing something is (see Sentencia De anima [SdA], iii, lc. 11, no. 746); and judgment (sometimes called composition and division), by which a person comes to know that what he has apprehended actually exists (or not) in reality (SdA, iii, lc. 11, no. 747).

For example, upon seeing a pine tree (say), a person will understand first what sort of thing it is (namely, a pine tree), and then with a second action will see that it does, in fact, exist in reality. (These actions can occur simultaneously in time, but the apprehension is logically prior to the judgment.)

Hence, when the Catechism says that conscience is a judgment, it means that it is an act of the human intellect of the second kind: it affirms (or denies) the existence of something in reality. The existence of what? Of the moral rightness (or wrongness) of a past, present, or future action.

In other words, it affirms, “This particular action that I propose to do (or am doing now, or have done in the past) is right (or wrong).” Or more specifically, “This action that I am about to do (or this action that I am refraining from doing) is obligatory (or prohibited) for me.”

The habits of the intellect that help the conscience

How does the intellect arrive at this judgment? As with any other use of the intellect, it uses its prior knowledge to make an inference.

The intellect cannot make this judgment (at least, not make it correctly), however, without the help of three habits, or stable dispositions:

1. The synderesis, or the habit that grants us knowledge of the first and highest moral principles: for example, the principle by which we know that we are obligated to choose the good and avoid evil. This habit is universal to all men, without exception, because it is present in every moral action whatsoever (even if the person chooses to act evilly—i.e, he sins; he can only sin if he knows that he is choosing something evil at the expense of the good). (See S.Th. Ia, q. 79, a. 12; also CCC 1780.)

2. What could be called moral science (“science” in the classical sense), or moral knowledge, by which we come to know (through personal reflection, the help of society, and—especially for Catholics—the help of the Church) the general principles by which we are to guide our actions. For example, moral science teaches us that human life is always to be respected; that sexual relations are reserved for marriage alone; and so forth. (This, as we will see, is what constitutes the formation of conscience; see CCC 1783-1785.)

3. Prudence, or moral wisdom, which is the habit that helps us apply the general principles to particular situations. (For example, it is prudence that helps a police officer to decide whether, in this situation, using lethal force is licit and still in keeping with the general principle that human life is always to be respected; also, it is prudence that helps a person decide that watching this particular movie, say, would be detrimental to the practice of chastity.) (See CCC 1806; also, also S.Th. IIa-IIae, qq. 46-47 on prudence.)

If these habits have been formed correctly, then they remain with us, even when the intellect is not is use (e.g., when we are sleeping). Whenever the intellect makes a judgment about a particular human action (past, preset or future), we call that conscience. Naturally, it will make much better judgments if it has properly formed moral science and prudence.

The erroneous conscience

The problem is that our intellect does not always judge correctly: we can be mistaken. For instance, if I see a tree at a large distance, I might mistake it for a pine tree, whereas it is really a spruce.

In a similar way, a person might think that a particular action is morally acceptable, when in fact it is morally wrong, and vice-versa. In this case, we are dealing with what is called an erroneous conscience. (See CCC 1790-1794.)

This helps us to make the following observations:

1. Man is absolutely obligated to follow the dictates of his conscience, even if it is erroneous. In other words, if he honestly thinks that action x is obligatory for him, then he must do action x; if he thinks that action y is prohibited, then he must not do y. (See CCC 1790.)
2. There is an objective moral order that transcends the subjective order. In other words, whether I think something is right or wrong does not affect whether in actual fact it is right or wrong.
3. Man has a grave duty to form his conscience, so as to avoid discrepancies between what he thinks is right and wrong, and what actually is so. Even though the person is not morally guilty for following an erroneous conscience, he will still experience the evil effects of doing actions that are objectively evil. “Forming one’s conscience” means honing two of the habits that I mentioned earlier: moral science, and prudence. (See CCC 1783-1785.)
4. General moral principles admit of no exceptions; the concrete applications of those principles often do. (See CCC 1787.) For instance, human life is always to be respected; there are, however, situations in which the best that can be done is the limitation of evil (e.g., self-defense; see CCC 2263).

Conscience and the marital act (regarding the O.P.’s question)

À propos to the O.P.’s question: that sexual relations are reserved for marriage alone is a general moral principle. (See CCC 2348-2350.) Hence, it admits of no exceptions whatsoever. (I.e., all of the actions mentioned in CCC 2351-2356 are intrinsically immoral; see also VS 67 regarding intrinsically immoral actions.) A person might conceivably be mistaken about that principle, and hence commit no sin when he violates it, but in that case his conscience would be erroneous, and he would still suffer the consequences of his objectively evil actions.

The idea that the conscience might “overrule” a general moral principle, especially one taught and affirmed by the Catholic Church, is, therefore, absurd. Someone who is mistaken about a moral principle might escape being culpable for his actions, but it would not make his actions right.

What might confuse people is that the application of those principles is seldom easy. There are no one-size-fits-all solutions (as we saw above in the example of self-defense). It tends to be easier, however, to apply negative precepts, such as “you shall not commit adultery,” since these effectively delineate the point beyond which it is not possible to respect the general principles involved.

For instance, a married couple clearly cannot have sexual relations outside of their marriage (not even with a partner through an attempted civil marriage); conversely, an unmarried person clearly cannot engage in sexual relations at all. That much involves no discernment at all.

However, whether today, here and now, a couple should engage in the marital act is a much more complex issue. Within a marriage, the marital act is, of course, generally licit—indeed holy and encouraged. But there could be situations that make it unadvisable, or even illicit (e.g., if one of the spouses is very ill or in pain, or if pregnancy would be dangerous or unduly burdensome). In those situations, careful discernment is warranted.