''Demogorgon, the prince of demons, wallows in filth and can spread a quickly fatal illness to his victims while rending them. He is a mighty spellcaster, and he can drain the life of mortals with a touch of his tail.''
Because Demogorgon's attacks make you deathly sick, he is generally regarded as one of the nastiest monsters in Nethack and Slash'EM. There are other monsters with sickness attacks, but they all have vulnerabilities which make them easier than Demogorgon. (Rot worms are slow, Juiblex can be defeated with a wand of digging, and Pestilence and Scorpius are vulnerable to death rays.)
Demogorgon has two sickness attacks per move. His first attack will leave you with between 20 and (19 + Con) turns to live. Subsequent sickness attacks will reduce your time to live by a factor of about three. To be precise, each attack will divide your time to live by 3 (rounding down) and then add 1. Even if you are very fast and unburdened, he will occasionally get two moves in a row against you, but you'll get two moves against him more often.
Two consecutive moves from Demogorgon (four attacks) will always leave you with two turns to live, or one turn if the turn ends after Demogorgon's move. Here's how it works:
Demogorgon stings! Your time to live is now 20.
Demogorgon stings! Your ttl = 7.
Demogorgon stings! Your ttl = 3.
Demogorgon stings! Your ttl = 2.
The initial ttl can be as much as 39, and perhaps the turn ended after Demogorgon's second sting and your ttl went down by one turn at that point (and any attempt you made to heal yourself failed). Even so, you will still have exactly two turns to live after the fourth sting. Any more stings will knock your ttl down to 1, and you will die at the end of the current turn, although you may get one move during this turn in which to heal yourself.
What all this amounts to, assuming that you are very fast and unburdened, is that if whenever you are ill, you cast cure sickness at 0% failure or use some other 100% reliable method of healing yourself, then Demogorgon will never kill you with sickness, and you will be faster than him, so you will sometimes get to take a swing at him. It is also safe to attack if Demogorgon had just one move this round and will get just one move next round, but only experts will be able to calculate this. Any other strategy runs some risk of death, so wear an amulet of life saving.
In vanilla Nethack, Demogorgon only appears when something else summons him, and many people have had several ascensions without ever encountering him. In Slash'EM, he has his own level in Gehennom, so you will have to face him. This spoiler was written for Slash'EM, but parts of it may also be useful for vanilla Nethack.
It is possible to turn Demogorgon to stone by hitting him with a cockatrice corpse, throwing a cockatrice egg at him, or polymorphing yourself into a petrifying monster (preferably a tough one such as a basilisk) and letting him sting you. Some people wish for a cockatrice corpse once they realise they are on his level. If you're out of wishes and do not have polymorph control, sometimes he will summon a cockatrice himself.
Demogorgon can be paralysed with a potion of paralysis, but make sure you are wearing a ring of free action. Don't throw the potion, as it will very likely miss. Instead, wield it against him to guarantee a hit.
You can also polymorph into a black dragon and disintegrate him, or use a ring of conflict to get your pet black dragon to breathe disintegration at him.
If you happen to have Houchou (an artifact spoon of insta-destruction), you could throw it at Demogorgon.
Demogorgon respects the mystic word Elbereth, so you could engrave it on the up stairs with a wand of fire or lightning. If you use any other engraving tool, the Elbereth will not be permanent and you will have to levitate above it to avoid smudging it. See Kate Nepveu's Elbereth FAQ for more information about Elbereth. Standing on a scroll of scare monster will also protect you, and you won't need to worry about eroding it.
A potion of invulnerability is useless against Demogorgon; it does not protect you from becoming deathly sick or dying from your illness. [Is this intentional?]
On the subject of strategies that don't work, Demogorgon is completely untameable, has such high magic resistance and level that it is impossible to pacify him with charm monster, and cannot be summoned by sacrificing your own race.
To defeat him in combat, you'll have to stop him teleporting away to heal himself. The way to do this is known as the upstairs/downstairs trick, and is also useful against other named demons and quest nemeses. When they teleport away, they teleport to the up staircase, and if you're on the up staircase, they don't teleport away from you. But when you meet them, you'll be much closer to the down staircase. Fortunately, if they're next to you when you go upstairs or downstairs, they will follow you.
So the thing to do is head for the down staircase, wait until Demogorgon's next to you, and head down. (Repeatedly healing yourself during this whole procedure, of course.) If you land on the up staircase, you're all set to fight him. If he lands there, fight him anyway until he either moves away and lets you in, or escapes upstairs (in which case you follow him and start it all again). Once you're on the up staircase, stay there until you've killed Demogorgon.
While fighting him, you need to keep switching between attacking him and healing yourself. The safe thing to do is heal yourself whenever you are ill and attack only when you're not - as long as you are very fast and unburdened, you will occasionally get two moves in a row, giving you a chance to attack. If you like living dangerously, get into a whack/heal cycle - heal yourself and take a swing at him on alternate moves. (Historical note: in Nethack 3.2.2 and versions of Slash'EM based on it, a very fast and unburdened character was precisely twice as fast as Demogorgon, so this tactic was perfectly safe.)
Clearly, you want to kill Demogorgon as quickly as possible to minimise the chance of YASD. It may help to know that like all demons he is extra vulnerable to silver weapons, and that his only resistances are to fire and poison.
The best method of healing yourself is to cast the spell of cure sickness at 0% failure. For some classes (e.g. Wizards trying to kill Demogorgon with Magicbane) it is wise to be at least Skilled in healing spells, so that you can keep casting cure sickness when you're out of magical energy.
If you're not a spellcaster, you can also cure yourself by drinking blessed potions of healing, non-cursed potions of extra or full healing, or holy water (non-chaotic characters only). Polymorphing into a new version of yourself will cure sickness; this is probably most useful for Doppelgangers, who can use the #youpoly command. You can use a blessed +6 or +7 unicorn horn, but it has a 10% failure rate. Eucalyptus leaves are another way of curing sickness, but I wouldn't recommend them - they might be rotten, and you do not want to be unconscious with Demogorgon around.