Newman discussed conscience in many places; if you type "conscience" into the fine search-engine at newmanreader.org, you will be given 547 places in which the word appears! Two of the places where his mature thought on it are to be found are chapter 5 of the Grammar of Assent, where it provides an argument for the existence of God, and chapter 5 of the Letter to the Duke of Norfolk, which concludes with the famous passage that seems to be all that some people know of Newmans writings:

I add one remark. Certainly, if I am obliged to bring religion into after-dinner toasts, (which indeed does not seem quite the thing) I shall drinkto the Pope, if you please,still, to Conscience first, and to the Pope afterwards.

In this chapter Newman responds to the accusation made by William Gladstone that the definitions of the first Vatican Council had given the pope absolute authority and required Catholics to forfeit their intellectual and moral freedom. In the previous chapter Newman had addressed the question of the"double allegiance" by which Catholics are said to be bound. Newman there adduced cases where one should obey the pope rather than the civil authority, and others where one should side with the civil power and not with the pope. He wrote:

When, then, Mr. Gladstone asks Catholics how they can obey the Queen and yet obey the Pope, since it may happen that the commands of the two authorities may clash, I answer, that it is my rule, both to obey the one and to obey the other, but that there is no rule in this world without exceptions, and if either the Pope or the Queen demanded of me an "Absolute Obedience," he or she would be transgressing the laws of human society. I give an absolute obedience to neither. Further, if ever this double allegiance pulled me in contrary ways, which in this age of the world I think it never will, then I should decide according to the particular case, which is beyond all rule, and must be decided on its own merits. I should look to see what theologians could do for me, what the Bishops and clergy around me, what my confessor; what friends whom I revered: and if, after all, I could not take their view of the matter, then I must rule myself by my own judgment and my own conscience. But all this is hypothetical and unreal.Here, of course, it will be objected to me, that I am, after all, having recourse to the Protestant doctrine of Private Judgment; not so; it is the Protestant doctrine that Private Judgment is our ordinary guide in religious matters, but I use it, in the case in question, in very extraordinary and rare, nay, impossible emergencies. Do not the highest Tories thus defend the substitution of William for James II.? It is a great mistake to suppose our state in the Catholic Church is so entirely subjected to rule and system, that we are never thrown upon what is called by divines "the Providence of God."

To clarify things, Newman then provided his chapter on conscience. It begins by relatingconscience to the moral character of God:

I say, then, that the Supreme Being is of a certain character, which, expressed in human language, we call ethical. He has the attributes of justice, truth, wisdom, sanctity, benevolence and mercy, as eternal characteristics in His nature, the very Law of His being, identical with Himself; and next, when He became Creator, He implanted this Law, which is Himself, in the intelligence of all His rational creatures. The Divine Law, then, is the rule of ethical truth, the standard of right and wrong, a sovereign, irreversible, absolute authority in the presence of men and Angels. "The eternal law," says St. Augustine, "is the Divine Reason or Will of God, commanding the observance, forbidding the disturbance, of the natural order of things." "The natural law," says St. Thomas, "is an impression of the Divine Light in us, a participation of the eternal law in the rational creature." ... This law, as apprehended in the minds of individual men, is called "conscience;" and though it may suffer refraction in passing into the intellectual medium of each, it is not therefore so affected as to lose its character of being the Divine Law, but still has, as such, the prerogative of commanding obedience.

Associating conscience with the eternal law of God, Newman hastens to clarify what he does and doesnot mean by it:

This view of conscience, I know, is very different from that ordinarily taken of it, both by the science and literature, and by the public opinion, of this day. It is founded on the doctrine that conscience is the voice of God, whereas it is fashionable on all hands now to consider it in one way or another a creation of man. Of course, there are great and broad exceptions to this statement. It is not true of many or most religious bodies of men; especially not of their teachers and ministers. When Anglicans, Wesleyans, the various Presbyterian sects in Scotland, and other denominations among us, speak of conscience, they mean what we mean, the voice of God in the nature and heart of man, as distinct from the voice of Revelation. They speak of a principle planted within us, before we have had any training, although training and experience are necessary for its strength, growth, and due formation. They consider it a constituent element of the mind, as our perception of other ideas may be, as our powers of reasoning, as our sense of order and the beautiful, and our other intellectual endowments. They consider it, as Catholics consider it, to be the internal witness of both the existence and the law of God. They think it holds of God, and not of man, as an Angel walking on the earth would be no citizen or dependent of the Civil Power. They would not allow, any more than we do, that it could be resolved into any combination of principles in our nature, more elementary than itself; nay, though it may be called, and is, a law of the mind, they would not grant that it was nothing more; I mean, that it was not a dictate, nor conveyed the notion of responsibility, of duty, of a threat and a promise, with a vividness which discriminated it from all other constituents of our nature.This, at least, is how I read the doctrine of Protestants as well as of Catholics. The rule and measure of duty is not utility, nor expedience, nor the happiness of the greatest number, nor State convenience, nor fitness, order, and the pulchrum. Conscience is not a long-sighted selfishness, nor a desire to be consistent with oneself; but it is a messenger from Him, who, both in nature and in grace, speaks to us behind a veil, and teaches and rules us by His representatives. Conscience is the aboriginal Vicar of Christ, a prophet in its informations, a monarch in its peremptoriness, a priest in its blessings and anathemas, and, even though the eternal priesthood throughout the Church could cease to be, in it the sacerdotal principle would remain and would have a sway.

Newman then observes that philosophers for some time have sought to undo this traditional notion:

We are told that conscience is but a twist in primitive and untutored man; that its dictate is an imagination; that the very notion of guiltiness, which that dictate enforces, is simply irrational, for how can there possibly be freedom of will, how can there be consequent responsibility, in that infinite eternal network of cause and effect, in which we helplessly lie? and what retribution have we to fear, when we have had no real choice to do good or evil?

And then there is the popular misunderstanding of conscience:

There, no more than in the intellectual world, does "conscience" retain the old, true, Catholic meaning of the word. There too the idea, the presence of a Moral Governor is far away from the use of it, frequent and emphatic as that use of it is. When men advocate the rights of conscience, they in no sense mean the rights of the Creator, nor the duty to Him, in thought and deed, of the creature; but the right of thinking, speaking, writing, and acting, according to their judgment or their humour, without any thought of God at all. They do not even pretend to go by any moral rule, but they demand, what they think is an Englishman's prerogative, for each to be his own master in all things, and to profess what he pleases, asking no one's leave, and accounting priest or preacher, speaker or writer, unutterably impertinent, who dares to say a word against his going to perdition, if he like it, in his own way. Conscience has rights because it has duties; but in this age, with a large portion of the public, it is the very right and freedom of conscience to dispense with conscience, to ignore a Lawgiver and Judge, to be independent of unseen obligations. It becomes a licence to take up any or no religion, to take up this or that and let it go again, to go to church, to go to chapel, to boast of being above all religions and to be an impartial critic of each of them. Conscience is a stern monitor, but in this century it has been superseded by a counterfeit, which the eighteen centuries prior to it never heard of, and could not have mistaken for it, if they had. It is the right of self-will.

(Later in the chapter Newman distinguishes conscience from "a fancy or an opinion"; instead it is "a dutiful obedience to what claims to be a divine voice, speaking within us.")Nineteenth-century papal condemnations of "liberty of conscience," Newman argued,were aimed at this popular misunderstanding of conscience. "Both Popes certainly scoff at the so-called "liberty of conscience", but there is no scoffing of any Pope, in formal documents addressed to the faithful at large, at that most serious doctrine, the right and the duty of following that Divine Authority, the voice of conscience, on which in truth the Church herself is built."To explain that last sentence, he offers this:

So indeed it is; did the Pope speak against Conscience in the true sense of the word, he would commit a suicidal act. He would be cutting the ground from under his feet. His very mission is to proclaim the moral law, and to protect and strengthen that "Light which enlighteneth every man that cometh into the world." On the law of conscience and its sacredness are founded both his authority in theory and his power in fact. Whether this or that particular Pope in this bad world always kept this great truth in view in all he did, it is for history to tell. I am considering here the Papacy in its office and its duties, and in reference to those who acknowledge its claims. They are not bound by the Pope's personal character or private acts, but by his formal teaching. Thus viewing his position, we shall find that it is by the universal sense of right and wrong, the consciousness of transgression, the pangs of guilt, and the dread of retribution, as first principles deeply lodged in the hearts of men, it is thus and only thus, that he has gained his footing in the world and achieved his success. It is his claim to come from the Divine Lawgiver, in order to elicit, protect, and enforce those truths which the Lawgiver has sown in our very nature, it is this and this only that is the explanation of his length of life more than antediluvian. The championship of the Moral Law and of conscience is his raison d'tre. The fact of his mission is the answer to the complaints of those who feel the insufficiency of the natural light; and the insufficiency of that light is the justification of his mission.

Conscience is a very fragile plant:

All sciences, except the science of Religion, have their certainty in themselves; as far as they are sciences, they consist of necessary conclusions from undeniable premises, or of phenomena manipulated into general truths by an irresistible induction. But the sense of right and wrong, which is the first element in religion, is so delicate, so fitful, so easily puzzled, obscured, perverted, so subtle in its argumentative methods, so impressible by education, so biassed by pride and passion, so unsteady in its course, that, in the struggle for existence amid the various exercises and triumphs of the human intellect, this sense is at once the highest of all teachers, yet the least luminous; and the Church, the Pope, the Hierarchy are, in the Divine purpose, the supply of an urgent demand. Natural Religion, certain as are its grounds and its doctrines as addressed to thoughtful, serious minds, needs, in order that it may speak to mankind with effect and subdue the world, to be sustained and completed by Revelation.

Newman then comes to the possibility of conflict between conscience and pope. He clarifies that in the cases he is considering, "conscience is not a judgment upon any speculative truth, any abstract doctrine, but bears immediately on conduct, on something to be done or not done.... Hence conscience cannot come into direct collision with the Church's or the Pope's infallibility; which is engaged in general propositions, and in the condemnation of particular and given errors." He is not, then, talking about conscientious dissent on points of doctrine. He then goes on:

Next, I observe that, conscience being a practical dictate, a collision is possible between it and the Pope's authority only when the Pope legislates, or gives particular orders, and the like. But a Pope is not infallible in his laws, nor in his commands, nor in his acts of state, nor in his administration, nor in his public policy. Let it be observed that the Vatican Council has left him just as it found him here.

Gladstone had misinterpreted Vatican I on the point. Infallibility does not affect disciplinary measures:

What have excommunication and interdict to do with Infallibility? Was St. Peter infallible on that occasion at Antioch when St. Paul withstood him? was St. Victor infallible when he separated from his communion the Asiatic Churches? or Liberius when in like manner he excommunicated Athanasius? And, to come to later times, was Gregory XIII., when he had a medal struck in honour of the Bartholomew massacre? or Paul IV. in his conduct towards Elizabeth? or Sextus V. when he blessed the Armada? or Urban VIII. when he persecuted Galileo? No Catholic ever pretends that these Popes were infallible in these acts. Since then infallibility alone could block the exercise of conscience, and the Pope is not infallible in that subject-matter in which conscience is of supreme authority, no deadlock, such as is implied in the objection which I am answering, can take place between conscience and the Pope.

Which brings him to his conclusion, which he buttresses with references to traditional theological authorities:

But, of course, I have to say again, lest I should be misunderstood, that when I speak of Conscience, I mean conscience truly so called. When it has the right of opposing the supreme, though not infallible Authority of the Pope, it must be something more than that miserable counterfeit which, as I have said above, now goes by the name. If in a particular case it is to be taken as a sacred and sovereign monitor, its dictate, in order to prevail against the voice of the Pope, must follow upon serious thought, prayer, and all available means of arriving at a right judgment on the matter in question. And further, obedience to the Pope is what is called "in possession;" that is, the onus probandi of establishing a case against him lies, as in all cases of exception, on the side of conscience. Unless a man is able to say to himself, as in the Presence of God, that he must not, and dare not, act upon the Papal injunction, he is bound to obey it, and would commit a great sin in disobeying it. Prim facie it is his bounden duty, even from a sentiment of loyalty, to believe the Pope right and to act accordingly. He must vanquish that mean, ungenerous, selfish, vulgar spirit of his nature, which, at the very first rumour of a command, places itself in opposition to the Superior who gives it, asks itself whether he is not exceeding his right, and rejoices, in a moral and practical matter to commence with scepticism. He must have no wilful determination to exercise a right of thinking, saying, doing just what he pleases, the question of truth and falsehood, right and wrong, the duty if possible of obedience, the love of speaking as his Head speaks, and of standing in all cases on his Head's side, being simply discarded. If this necessary rule were observed, collisions between the Pope's authority and the authority of conscience would be very rare. On the other hand, in the fact that, after all, in extraordinary cases, the conscience of each individual is free, we have a safeguard and security, were security necessary (which is a most gratuitous supposition), that no Pope ever will be able, as the objection supposes, to create a false conscience for his own ends.

Two months after the Letter to the Duke of Norfolk was published, Newman explained in a letter what he had attempted:

As to the question of conscience, neither Gladstone nor any other Protestant would deny that law must give way before it, when it is the personal suggestion of the individual and dictates to act thus or that. What they accuse Catholics of is, of in every case ruling our conscience by the command of the Pope or of our Confessor, so that conscience ceases to be a tribunal within us, and to fall under the notion of a natural right of man. Instead then of being what we mean by conscience, they say, it is nothing else than the command of a political party, external to England, which makes use of a sacred name for its own purposes. This is the representation which I had to meet, and I think have met in my letterby showing that after all, whatever our duties to the Pope, there are cases in which conscience may come into collision with his word, and in which conscience must be followed, not he" Letters and Diaries, vol XXVII, p. 244).