How binding are encyclicals on practicing Catholic's beliefs?

Encyclicals are inerrant; otherwise, the Church would demand the faithful to assent to what might contain error, and the Church cannot lead one into error because the Church is indefectible. From Pope Pius XII's Humani Generis:

20. Nor must it be thought that what is expounded in Encyclical Letters does not of itself demand consent, since in writing such Letters the Popes do not exercise the supreme power of their Teaching Authority. For these matters are taught with the ordinary teaching authority, of which it is true to say: "He who heareth you, heareth me";[Luke, X, 16] and generally what is expounded and inculcated in Encyclical Letters already for other reasons appertains to Catholic doctrine. But if the Supreme Pontiffs in their official documents purposely pass judgment on a matter up to that time under dispute, it is obvious that that matter, according to the mind and will of the Pontiffs, cannot be any longer considered a question open to discussion among theologians.

Encyclicals can even contain ex cathedra (infallible) dogmas, such as Pope Pius XI's condemnation of contraception in his 1930 encyclical on marriage, Casti Connubii.

The scope of obedience due to the popes and their encyclicals is such that Pope St. Pius X was compelled to say this in his Address to the Priests of the Apostolic Union (Nov. 18, 1912) included in the official Acta Apostolicae Sedis 4 [1912], p. 695):

When one loves the pope one does not stop to debate about what he advises or demands, to ask how far the rigorous duty of obedience extends and to mark the limit of this obligation. When one loves the pope, one does not object that he has not spoken clearly enough, as if he were obliged to repeat into the ear of each individual his will, so often clearly expressed, not only viva voce, but also by letters and other public documents; one does not call his orders into doubt on the pretext – easily advanced by whoever does not wish to obey – that they emanate not directly from him, but from his entourage; one does not limit the field in which he can and should exercise his will; one does not oppose to the authority of the pope that of other persons, however learned, who differ in opinion from the pope. Besides, however great their knowledge, their holiness is wanting, for there can be no holiness where there is disagreement with the pope.

The most succinct explanation of a Catholic's reception of encyclicals is expressed in Humani Generis by Pope Pius XII:

Nor must it be thought that what is expounded in Encyclical Letters does not of itself demand consent, since in writing such Letters the Popes do not exercise the supreme power of their Teaching Authority. For these matters are taught with the ordinary teaching authority, of which it is true to say: “He who heareth you, heareth me” [Lk 10:16]; and generally what is expounded and inculcated in Encyclical Letters already for other reasons appertains to Catholic doctrine. But if the Supreme Pontiffs in their official documents purposely pass judgment on a matter up to that time under dispute, it is obvious that that matter, according to the mind and will of the Pontiffs, cannot be any longer considered a question open to discussion among theologians.

Pope Pius XI echoes the same sentiment in Casti Connubii, n. 104:

Wherefore, let the faithful also be on their guard against the overrated independence of private judgment and that false autonomy of human reason. For it is quite foreign to everyone bearing the name of a Christian to trust his own mental powers with such pride as to agree only with those things which he can examine from their inner nature, and to imagine that the Church, sent by God to teach and guide all nations, is not conversant with present affairs and circumstances; or even that they must obey only in those matters which she has decreed by solemn definition as though her other decisions might be presumed to be false or putting forward insufficient motive for truth and honesty. Quite to the contrary, a characteristic of all true followers of Christ, lettered or unlettered, is to suffer themselves to be guided and led in all things that touch upon faith or morals by the Holy Church of God through its Supreme Pastor the Roman Pontiff, who is himself guided by Jesus Christ Our Lord.

There are hundreds of similar quotes on papal infallibility, private judgment, obedience and the like. Let an explanation by Fr. Joachim Salaverri in Sacrae Theologiae Summa IB: On the Church of Christ (1955) suffice:

Authentic magisterium (from [Greek] authentia = authority) is the office of handing on doctrine instituted by a legitimate authority. Therefore, it implies in the teacher the power and office of handing on doctrine; but in the disciples [i.e. in the taught] the obligation and right to receive instruction. Magisterium can be authentic in two ways: in the broad sense and in the strict sense.

Authentic magisterium in the broad sense is that which by itself does not have the power to demand from the disciple the assent of the intellect. Such is, for example, the magisterium of a professor in a university. Authentic magisterium in the strict sense is that which has such power in itself to impose doctrine, that the disciples by that very fact are bound to give the assent of the intellect, because of the authority of the legate of God which the teacher makes use of.

Here we see that obedience is due in virtue of the authority of the teacher and not only in his infallible charisma.

Considering the strictness of the obedience due to the authentic magisterium, one might be compelled to ask: "What if the magisterium is wrong?" The answer, of course, is that the magisterium cannot be wrong in matters that pertain to salvation, and, even if it is wrong in some minor detail, a Catholic is perfectly safe in following the legitimate teaching authority, that is, he has moral certitude in following the teaching of the Church.

If a presumed Pope or council propose something manifestly contrary to revealed truth as their authentic teaching, they must be regarded as having lost all authority the minute they became heretics, and therefore are owed no obedience. This is confirmed by the below texts and many more, such as Cum Ex Apostolatus, Mystici Corporis, Pastor Aeternus, etc.:

Si Papa [1198], Pope Innocent III

“The Pope should not flatter himself about his power nor should he rashly glory in his honor and high estate, because the less he is judged by man, the more he is judged by God. Still the less can the Roman Pontiff glory because he can be judged by men, or rather, can be shown to be already judged, if for example he should wither away into heresy; because he who does not believe is already judged. In such a case it should be said of him: 'If salt should lose its savor, it is good for nothing but to be cast out and trampled under foot by men.’”

Institutiones Juris Canonici [1950] - Coronata

“If indeed such a situation would happen, he [the Roman Pontiff] would, by divine law, fall from office without any sentence, indeed, without even a declaratory one. He who openly professes heresy places himself outside the Church, and it is not likely that Christ would preserve the Primacy of His Church in one so unworthy. Wherefore, if the Roman Pontiff were to profess heresy, before any condemnatory sentence (which would be impossible anyway) he would lose his authority.”

St. Robert Bellarmine [1610]

“A Pope who is a manifest heretic automatically ceases to be a Pope and head, just as he ceases automatically to be a Christian and a member of the Church.”

In conclusion, a Catholic should not trouble himself with finding faults in Encyclicals or disagreeing with them, but rather with fulfilling the will of the Church the best he can, recognizing his own inability and imprudence.

How binding are encyclicals on practicing Catholic's beliefs?

It will depend on certain factors involved in a particular encyclical. If the pope is providing teachings on faith or morals, than the particular encyclical would be morally binding. Thus Humanae Vitae is obviously very morally binding on Catholics.

As for the binding force of these documents it is generally admitted that the mere fact that the pope should have given to any of his utterances the form of an encyclical does not necessarily constitute it an ex-cathedra pronouncement and invest it with infallible authority. The degree in which the infallible magisterium of the Holy See is committed must be judged from the circumstances, and from the language used in the particular case. In the early centuries the term encyclical was applied, not only to papal letters, but to certain letters emanating from bishops or archbishops and directed to their own flocks or to other bishops. Such letters addressed by a bishop to all his subjects in general are now commonly called pastorals. Amongst Anglicans, however, the name encyclical has recently been revived and applied, in imitation of papal usage, to circular letters issued by the English primates. Thus the reply of the Archbishops of Canterbury and York to the papal condemnation of Anglican Orders (this condemnation, "Apostolicæ Curæ", took the form of a Bull) was styled by its authors the Encyclical "Sæpius officio". - Encyclical

In any case, Catholic encyclicals should be be received in the light of the Catholic faith and should be seriously taken for the insights the popes are expressing from an viewpoint of the Catholic faith.

Encyclicals should be embraced in a positive light and taken seriously, even if no moral issues are brought up in the papal document.

One area that I had assumed was clearly in the optional category was papal social teaching. But at Evangelical Catholicism, M.J. Andrew argues that’s the latest encyclical is binding on all Catholics:

The error that both Leo XIII and Pius X are correcting is one that is still made today in Catholic circles, namely, that the social teaching of the Church is in someway optional, non-binding, and/or merely prudential. Indeed, in many respects our inclinations and political proclivities, which often are merely products of our locale and social environment, are met by a powerful counterweight in Catholic social teaching. And rightly so. If, as the two aforementioned popes assert, our social life (i.e., family, political, and economic activity) are primarily religious and moral in nature, then the Church, by virtue of her authority in matters of faith and morals, is our touchstone for learning how to conduct that social life . . . .

Yet, we still encounter the stubbornly persistent opinion among Catholics that the Church’s social doctrine is not binding–and if it is authoritative, then it is not as important or consequential as doctrines of faith. But this position is certainly not to be found in Catholic teaching. Indeed, it is simply a pernicious prejudice.

Andrew proffers an intriguing argument (persuasive enough to convince this Protestant). But is he right? If so, why do so many Catholics ignore—much less dismiss—this encyclical?

Are Social Encyclicals Binding on Catholics?