Tanar'ri
These are demons, the chaotic evil supernatural badguys, in 2nd Edition Dungeons and Dragons. After TSR got their new CEO Lorraine Williams and Gary Gygax left in disgust, the company decided that nutcases like Jack Chick and Patricia Pulling had to be pandered to, so Williams said "no more demons, or any of that junk". In 3e they wisely decided to use "demon" again, but kept the term "tanar'ri" as a subtype of demon. Since TSR bothered to register "tanar'ri" as a trademark, you will not find this demon subtype in Open Gaming License materials like the D20SRD.
Fiendish Codex II - Tyrants Of The Nine Hells states that Devils call all demons Tanar'ri for short, which is strange considering that in the same book it says that in the language of the devils "only one correct way exists to construct any given statement in Infernal." It also states that devils "except when in disguise, they find the urge to correct errors in spoken or written Infernal nearly impossible to resist." This may be a hint as to the origin of Lorraine Williams, since she also insisted on using the term "tanar'ri" instead of "demon."
Demon Lords
The first amongst the demons, the lords of the Tanar'ri (more commonly known as Demon Lords) are some of the most powerful, dangerous, cunning and deadly beings in existence. They attained their position by strength of arms and cleverness (mostly the former) to rule over their servants. Those who wish to challenge their lords are quickly struck down as an example and to keep the other minions in line.
Baphomet
Baphomet, the lord of the minotaurs, is a massive bull-headed brute. He revels in combat, but prefers to put his powerful-looking opponents into a magical maze and trample the smaller ones before crushing the big ones. His most common followers include minotaurs, bulezau and goristro demons, fiendish or half-fiendish minotaurs and Awakened animals such as dire animals or dinosaurs.
Baphomet's cult is on the rise, often finding worship in rural areas by evil rangers preaching the gospel of a nature deity of vengeance that will deliver the common folk from the evils of city-dwellers, eventually swaying them to his cause.
Baphomet lives in the Endless Maze, the 600th layer of the Abyss. He shares this layer with the Obyrith Pale Night, with whom he has some kind of understanding: neither of them encroach on each other's territories and they will help each other when one of them is invaded by one of their foes.
Demogorgon
The current Prince of Demons. He has the shape of a two-headed babboon with tentacles instead of arms. His left head is called Aameul and can warm any target to its way, while the right head Hethradiah can cause insanity. Demogorgon is a fearsome foe in battle, capable of rotting flesh from bones with but a caress of its tentacles and can drain the energy of anyone slapped by its tail.
Demogorgon has few followers amongst the humanoid races, but he has many amongst the more monstrous people. Yuan-ti, troglodytes, ixitxachitl and hosts of demons worship Demogorgon as their lord. These cults are depraved and engage in all manner of horrific rituals and sacrifices to their lord. Humanoids rarely worship him as cults, but individuals dedicated to the Prince of Demons tend to be just as dangerous, given that they are quite often high-ranking members of society.
Demogorgon's layer of The Gaping Maw is the 88th of the Abyss. While it's a tropical realm with plenty of water it is not a hospitable plane, with many different kinds of demons roving the surface and the ocean looking for things to devour. Demogorgon runs the great city of Lemoriax that is one of the premier slave markets in the Great Wheel. He frequently stalks through his city to whip his followers into a frenzy. His layer is linked to the next layer, Dagon's home of Shadowsea. The two are allied, which effectively means they stay off of each other's backs.
Fraz-Urb'luu
The Prince of Deception, Fraz-Urb'luu is a great pale plue winged monster with surprising magical ability. He is the patron of evil illusionists and tricksters, so powerful that he can easily summon demons and even other demon lords, though he prefers to not do the latter all that much because does not want to pis Demogorgon off.
Fraz-Urb'luu was one of Zagig's prisoners below Castle Greyhawk, trapped in a bas-relief for more than two centuries. When he was finally freed by a bunch of adventurers, whom he repaid by wrecking their gear and casting them into the Abyss. What a dick. Back in the 176th layer of the Abyss, Hollow's Heart, he spent time rebuilding his realm and searching for his magic staff. If both goals were to be achieved he would visit his vengance upon the world of man.
Cults of Deception pose as cults of exiting but obscure deities and try to convert people to their cause... only to be sacrificed to the Prince of Deception. His followers include large numbers of other deceptive dmons, including succubi, lilitu and a group of Chaotic Evil Rakshasa called the Hollow Rajas.
Graz'zt
The Dark Prince Graz'zt is something unusual amongst the princes of the Abyss. While the others all possess some degree of cleverness, at the end of the day they will use brute force to get what they want. Not so much in Graz'zt's case. He is skilled in the ways of seduction, guile, politics and plotting in ways that go beyond the abilities of even the most intelligent princes of hell. 4e suggested that this is because Graz'zt is an exiled devil, but this is the only version that posed this option.
He resmbles a tall, handsome humanoid with goat legs, horns, six fingered hands and yellow eyes betray him for what he really is. Graz'zt has a massive libido and counts many female priests, succubi, harpies and lamia amongst his followers, called the Chosen (with the suggestion he chooses them to join him in his bed). While he is of great intelligence he has met his match in a mortal, the witch-queen Iggwilv. She managed to trap him on the Material Plane to learn from the demon. The two of them eventually became lovers, culmulating in the birth of a child. Graz'zt slowly figured out that Iggwilv did not want to let him go back to his home. So he waited and plotted, eventually breaking free in a battle that costed Iggwilv most of her resources. But she escaped and was lured into the Abyss by the Dark Prince. She in turn was used by Graz'zt to regain his lost powers and grow stronger before she escaped. The two of them are currently in a mutual love-hate relationship, each trying to exploit the other to increase their power.
Graz'zt's realm is the largest in all of the Abyss. Covering the 45th, 46th and 47th layers of the Abyss, Azzagrat's primary attraction if the city of Zelatar, which somehow exists on all three layers at the same time. Each layer however hosts a different neighbourhood, and Graz'zt's citadel on the center layer.
Jubilex
The Faceless Lord Jubilex lords over all oozes, slimes and jellies. This is not a paricularly difficult task given the mindless nature of these creatures and Jubilex' lack of tact regarding such issues. He resmbles a massive puddle of green slime filled with eyes swimming around in it and pseudopods drippingwith ooze. His goals are simple: destroy everything that is not like him. He wants to kill and corrode everything in existence, destroying whatever it can touch with its oozing appendages. This makes him very unpopular with the other demon lords and all stay away from him as much as possible.
Jubilex has few cultists given that his charges are almost exclusively of low intelligence. When a deranged humanoid takes up the worship of the Faceless Lord it is quite possible that Jubilex doesn't really care about these worshippers, and even when an intelligent ooze such as Darkness Given Hunger, a Black Pudding possessed by a Hezrou, worships him Jubilex cares little.
Jubilex lives on the 222nd layer of the Abyss named Shedaklah. He contests for control of this plane with Zuggtmoy, the Queen of Fungi. While initially she had the upper hand since the events with Iuz and the Temple of Elemental Evil Jubilex has gained a lot of terrigory, currently battling with the forces of the Queen of Fungi and slowly gaining ground.
Malcanthet
Malcanthet, the Queen of the Succubi is a being of unparalleled beauty and cruelty. Taking the shape of an alluring succubus she prefers to let others do the fighting for her, either summoned minions or enslaving her attackers. Befitting her status of the Queen of the Succubi she really gets around: aside from her usual trysts she has Pazuzu and Demogorgon as her most frequent lovers, having sired several powerful succubi and unspeakable things with them respectively. She has had short affaris with most of the other demon lords, but these were mainly to get something she wanted. Yeenoghu is one of her enemies, the two of them still being embittered by the other foiling their plot to corrupt the same family of nobles, ending with the entire family being destroyed instead of corrupted. Her relationship with Graz'zt is even worse: both claim that the other turned them down and became embittered by this, and the two have been at war ever since. Her servants include aside from succubi and lilitus numbers of lamias, harpies and fiendish Nymphs.
Malcanthet lives on the 570th layer of the Abyss, a layer called Shendilavri. It is of unmatched beauty, surpassing even the heavens as a paradites. Here she lives in the city of Rivenheart, where all desires can be sated. Every imaginable perversion, debauchery and experience can be found here. Malcanthet's palace has no doors, and all are invited to enter and experience what lies within its forbidden halls. While thousands of people from all over the planes enter to find bliss within, few people leave.
Orcus
Looking like the forbidden lovechild of Khorne and Nagash, Orcus is the Prince of the Undead. He has his own page, which can be found here.
Yeenoghu
The King of the Gnolls, Yeenoghu is a savage beast. Despite not being the most powerful of the demon lords his savagery and that of his followers makes this demon lord a powerful opponent. This became clear when he and his forces invaded the layer above his own, the White Kingdom. This layer was ruled by Doresain, the King of the Ghouls. When his armies were destroyed Doresain swore fealty to Yeenoghu, forsaking his old master Orcus. Because of Orcus' recent plots his attention has been elsewhere, and it is as of yet unknown how he will reply to this treason.
Yeenoghu's greatest desire is to have the Material Plane inhabited by his chosen people, the gnolls. A world where his people are the dominant race, and all humans, elves, dwarves and halflings are either slaves or food. But he understands that gnolls alone are not numerous enough to achieve this goal, and as such he has to content with having non-human cults. One of the most powrful cults he ever had, originating from the Maure family was destroyed when Malcanthet also tried to corrupt the family, ending in the humans' destruction. They have resented each other ever since, with Yeenoghu sending frequent raids into Shendilavri to destroy it. These attacks have always been held at bay, but he hopes to one day destroy her and her realm. The King of the Gnolls also has a violent hatred for Baphomet, against whom he frequently wages war to little result.
Yeenoghu's realm is called exactly that, Yeenoghu's Realm. Located on the 422th layer of the Abyss it is covered in a great savannah where packs of feral gnolls, other hyena-like creatures and worse monsters make their home, preying on one another and the servants of Yeenoghu. The King of the Gnolls himself lives in a great palace drawn by thousands of slaves across the plane past various points of interest at a speed of about one year per round.
Zuggtmoy
Zuggtmoy, The Queen of Fungi is an ambitious but not very successful demon lord. Given her domain over fungi and mushrooms she has difficulty finding people to worship them. So she opts with having to disguise her cults, creating cover cults that her own followers kept in line so that they could stay in charge. The Temple of Elemental Evil was such an endeavour, only for it to backfire when the Circle of Eight destroyed the cult and imprisoned her. She managed to escape the Material Plane at the cost of a good deal of her essence, only to discover that her enemy Jubilex had conquered a good deal of her home layer.
Aside from her rivalry with Jubilex she has frequently been at odds with Lolth prior to her gaining significant power as a deity, putting her a class above the Queen of Fungi. On her home layer of Shedaklah she lives in a palace build from the largest fungi in the multiverse, rising more than four miles into the air. Here she conducts horrific experiments and plots to gain foothold on the Material Plane and to defeat the hated Jubilex.