When the Draenei were introduced as a playable race, a few players scratched their heads in confusion. While there were Draenei in Warcraft III: The Frozen Throne, they all had the look of what we'd now call "Lost Ones" (those found in Swamp of Sorrows and Zangarmarsh.) The Draenei in Burning Crusade, which we're all very familiar with at this point, looked instead like the Eredar, the super-powerful demons that, prior to that point, had only been seen in the form of Archimonde in WCIII Reign of Chaos, and Kil'jaeden in the expansion, Frozen Throne.
Blizzard explained this by showing that the Eredar had once been a mortal race that then got corrupted and transformed into these "Man'ari Eredar," and that the Draenei were the last remnant of the Eredar who had not been corrupted. They then further explained the difference in appearance by introducing the idea of the horrific Red Mist used by the Horde in the Siege of Shattrath (which in Warlords of Draenor, was hinted to be the blood of Sethe, modified to mutate Draenei instead of Arrakoa.)
The problem with this was that there was a big continuity error. In Warcraft III, the Eredar were given the backstory of being the demons whose evil acts caused Sargeras to despair and turn evil himself. That doesn't really work with the new continuity, as it wouldn't make a whole lot of sense for the Eredar and Sargeras to have effectively corrupted each other.
So there was a retcon (and frankly I'm glad there was, as the Draenei are possibly my favorite playable race, and they have a really cool backstory) which gave that role of "corrupters of Sargeras" to the Dreadlords, aka the Nathrezim.
The Nathrezim are certainly officer-class within the Burning Legion - you tend not to see armies composed of them, instead finding them in leadership positions. They're probably below Pit Lords (Annihilan) and the Eredar.
Dreadlords played a key role in the Third War. Tichondrius was the commander of the Legion forces (under Archimonde of course,) while Mal'ganis was in charge of keeping the Lich King on task. The Lich King engineered Arthas' victory and his corruption by allowing him to kill Mal'ganis. Mal'ganis would later come back in Wrath of the Lich King, but beyond simply appearing after we defeat his Scarlet Onslaught, he just goes away.
(In all honesty, this plot always frustrated me, as my take had always been that, because Mal'ganis had been slain with Frostmourne, his soul should have been trapped within the blade, meaning that yes, now that the sword had been shattered, he should be free, but he should not have been able to come back in 3.0.)
Three Dreadlords were in control of the Plaguelands after the Third War: Balnazzar, Mephistroph, and Varimathras. Balnazzar would possess the commander of the Scarlet Crusade, which is likely the reason it went all Spanish-Inquisition-level-crazy. Varimathras was defeated by Sylvanas Windrunner and pledged to serve her, though he ultimately betrayed her, causing the Wrathgate Incident. He was killed in the Undercity (just as Balnazzar had been killed in Stratholme,) but being a Dreadlord, it's certain that he's still out there.
We never dealt with Mephistroph, but it appears that some of the Artifact Weapon quests will involve him.
As clearly pre-Legion demons, it's unclear what the Nathrezim's origins were. We don't really know what exactly it means to be a demon, but, and this is speculation on my part, it seems that a humanoid race can become a demon by being infused with the energy of the Twisitng Nether. While typically this is done with specifically Fel energy, the Green Fire quest chain showed us that demons can also feed on pure Arcane energy, which is what Illidan was doing with his demonic minions using the Reliquary of Souls.
Some have speculated that the Nathrezim might have come from Xorroth, the world that Dreadsteeds (the Warlock epic mount) come from, though it's possible this connection is simply thanks to the prefix of "Dread" in the names. We know that Xorroth was a twin planet with another world called Xerrath (the passive you gain when you get Green Fire is the "Codex of Xerrath.) Xerrath was destroyed by demonic magic.
It's possible that all demons begin as some sort of natural creature - humanoids, beasts, or elementals (Voidwalkers, while classified as demons for mechanical reasons, are technically aberrations, coming from the Void.) The Titans certainly used demons - we know that Doomguards were created to hunt down people using Sacrificial Magic - which is why they are now summoned using that kind of magic (this used to be more explicit in game mechanics, as you'd actually need to sacrifice a party member to summon one.) It's perhaps a little ambiguous if Doomguards were technically demons when the Titans were using them, but it shows that a lot of demons were originally something else that were made that way.
It strikes me that the Dreadlords might have been the first demons. But that doesn't necessarily mean that they were always demons, eternally. If Xorroth was their home world, they may have been an advanced race, somewhat like the Eredar. Yet without the prodding by some Dark Titan, the Nathrezim might have drawn upon the vast power within the Twisting Nether, infusing their being with it and getting corrupted by the power they found.
Perhaps the Nathrezim were the ones to develop Fel Magic, twisting the morally neutral Arcane into something inherently destructive. Perhaps they destroyed their twin planet, and such a depraved and cruel act was what pushed Sargeras over the edge.