According to Catholic theology, what does the suffering of hell consist of?
If there a principal suffering, what is it?
1034 Jesus often speaks of "Gehenna" of "the unquenchable fire" reserved for those who to the end of their lives refuse to believe and be converted, where both soul and body can be lost. Jesus solemnly proclaims that he "will send his angels, and they will gather . . . all evil doers, and throw them into the furnace of fire," and that he will pronounce the condemnation: "Depart from me, you cursed, into the eternal fire!"
1035 The teaching of the Church affirms the existence of hell and its eternity. Immediately after death the souls of those who die in a state of mortal sin descend into hell, where they suffer the punishments of hell, "eternal fire." The chief punishment of hell is eternal separation from God, in whom alone man can possess the life and happiness for which he was created and for which he longs.
The emphasis in this passage is mine. The catechism says that separation from God is the chief punishment in hell. The reference to hell fire is placed in a strange context, given that the Church can hardly deny Jesus' words, yet it seems (to me) as if the Church would rather gloss over the hell fire aspect - perhaps because the concept of fire causing suffering to a pure spirit is not something easy to explain.
Apart from the chapter on Hell that appears in the Catholic Catechism (see the other answer), Saint Faustina Kowalska gives a detailed description of that spiritual state and its sufferings in her Diary, number 741 (you may find a copy here). She saw it for herself:
Today, I was led by an angel to the chasms of hell. It is a place of great torture; how awesomely large and extensive it is! The kinds of tortures I saw: the first torture that constitutes hell is the loss of God; the second is perpetual remorse of conscience; the third is that one's condition will never change; the fourth is the fire that will penetrate the soul without destroying it – a terrible suffering, since it is purely spiritual fire, lit by God's anger; the fifth torture is continual darkness and a terrible suffocating smell, and despite the darkness, the devils and the souls of the damned see each other and all the evil, both of others and their own; the sixth torture is the constant company of Satan; the seventh torture is horrible despair, hatred of God, vile words, curses and blasphemies. These are the tortures suffered by all the damned together, but that is not the end of the sufferings. There are special tortures destined for particular souls. These are the torments of the senses. Each soul undergoes terrible and indescribable sufferings, related to the manner in which it has sinned. There are caverns and pits of torture where one form of agony differs from another. I would have died at the very sight of these tortures if the omnipotence of God had not supported me. Let the sinner know that he will be tortured throughout all eternity, in those senses which he made use of to sin. I am writing this at the command of God, so that no soul may find an excuse by saying there is no hell, or that nobody has ever been there, and so no one can say what it is like.
I, Sister Faustina, by the order of God, have visited the abysses of hell so that I might tell souls about it and testify to its existence. I cannot speak about it now; but I have received a command from God to leave it in writing. The devils were full of hatred for me, but they had to obey me at the command of God. What I have written is but a pale shadow of the things I saw. But I noticed one thing: that most of the souls there are those who disbelieved that there is a hell. When I came to, I could hardly recover from the fright. How terribly souls suffer there! Consequently, I pray even more fervently for the conversion of sinners. I incessantly plead God's mercy upon them. O my Jesus, I would rather be in agony until the end of the world, amidst the greatest sufferings, than offend You by the least sin.
This vision, and many others in her Diary, rank as private revelations, but they are widely regarded as true in the Catholic world, like those of Fatima or Lourdes. Inspired by these revelations, Saint John Paul II designated the Sunday after Easter as the "Divine Mercy Sunday".
Edit: now I realise that in Fatima another description was given:
Well, the secret is made up of three distinct parts, two of which I am now going to reveal.
The first part is the vision of hell.
Our Lady showed us a great sea of fire which seemed to be under the earth. Plunged in this fire were demons and souls in human form, like transparent burning embers, all blackened or burnished bronze, floating about in the conflagration, now raised into the air by the flames that issued from within themselves together with great clouds of smoke, now falling back on every side like sparks in a huge fire, without weight or equilibrium, and amid shrieks and groans of pain and despair, which horrified us and made us tremble with fear. The demons could be distinguished by their terrifying and repulsive likeness to frightful and unknown animals, all black and transparent. This vision lasted but an instant. How can we ever be grateful enough to our kind heavenly Mother, who had already prepared us by promising, in the first Apparition, to take us to heaven. Otherwise, I think we would have died of fear and terror.
We then looked up at Our Lady, who said to us so kindly and so sadly:
“You have seen hell where the souls of poor sinners go. To save them, God wishes to establish in the world devotion to my Immaculate Heart".