Main races found in the Underdark
Drow. The Black skinned Elves, chosen race of Lolth.
Dark Elves. Elves who follow the dark paths, the progenitors of the Drow and former brethren to the surface Elves.
Norn. Ice Elves residing the Northern Wastes, formerly of Clan Mistrivven now devout followers of Dracos
Duergar. Deep, or Grey, Dwarves who have integrated themselves into the underdark, if not into Drow society.
Svirfneblin. Rarely seen offshoots of the Wee Folk, even the Dark Elves consider them dark and twisted creatures.
All of these are descended from the Elder races; elf or, in the case of the Dueregar, Dwarf.
Other known sentient Underdark races
Illithids. Known in legend as the Mind flayers. A power but now-lost race due to the Cataclysm of Fire eradicating them from existence.
Aztar. A Beast/Lizardkin species, recently released from centuries of entrapment under the psionic magic of Illithids.
Other Underdark inhabitants
Slave races and slavery in the Underdark is common place in the underdark, and many of the surface races can be found working as slaves, earning a living as mercenaries for hire, or as free men who would be tried as criminals upon the surface and chose a life in the darkness over their punishments upon the surface.
More information about these lesser races can be found here.