In AD&D cosmology, Tanar'ri (which is both singular and plural, much like the word "elk") are the paramortals commonly referred to by mortals as "demons." Whatever they are called, Tanar'ri are the incarnations of the chaotic evil alignment. Tanar'ri mainly dwell on the infinite layers of the Lower Plane known as Abyss.
The actual difference between a "Tanar'ri" and a "demon" is usually moot, but for sticklers and sages, any denizen of the Abyss may rightly be called a demon, but only certain types of demon are rightly classified as Tanar'ri. Unfortunately, the distinction between what is a Tanar'ri, and what is not is less clear than with their lawful counterparts, the Baatezu, or devils.
The known Tanar'ri species are:
Tanar'ri share a few abilities:
True children of chaos, Tanar'ri breeds take a variety of forms, although, being incarnations of evil, they tend to be pretty menacing. Some Tanar'ri are capable of mating with mortals, producing Alu-fiends if the mother is a Succubus, or Cambions if the mother is a mortal. When Tanar'ri want to create pure-breds, they simply perform their foul rituals on the souls of particularly evil mortals, which take the form of the giant larva that infest the Lower Planes, although the so-called "True" Tanar'ri are said to be born out of the raw essence of the Abyss itself, never having known a mortal's weaknesses. A Tanar'ri killed outside of the Abyss will reform in the Abyss after a given amount of time unless they were slain with a holy weapon, or the corpse was eaten (not reccomended for mortals).
Tanar'ri are famous for their supposedly infinite numbers, and their sociopathic tendancies. Advancement usually comes about by simply being able to physically take out your superiors.
The Tanar'ri have been embroiled in the Blood War with their lawful arch-nemisises, the Baatezu, or "devils" since the two races first met shortly after the dawn of time.
Bonus politics of role-playing games
In first-edition AD&D, the word "Tanar'ri" did not exist, and the creatures were simply called "demons." However, thanks to the combined legal efforts of philistines, by the time second-edition came out, the more ambiguous term, "Tanar'ri" was created. By 2000, when 3e was published, the D&D franchise had enough money, lawyers, and giant corporate owners that the game was allowed to exercise its 1st Amendment rights and put these beasties back in the "D" section of the Monsterous Manual, although "Tanar'ri" is still mentioned there, as their "proper name."
As an aside: in first-edition, the specific demon breeds had no names. They were simply "Type-I," "Type-II" and so forth. For second-edition, TSR pulled appropriate names from out of folklore and myth to attach to many of these supernatural critters, even as they were also assigning them a goofy group-name.