He’s the original, primeval god of the Underworld, presiding over the place of ultimate punishment.
One of the first gods to arise from the void of Creation, TARTARUS, like his siblings NYX and CHAOS, personifies ultimate formless gloom. Little is known of his personality, but as the first God of Hell we assume he is not a bucketful of laughs. Think keywords such as dour, depressing, demoralizing and dirge.
TARTARUS lives in the bowels of the Earth, many miles below as the anvil falls, and holds together the bottomless pits of the world. Presumably HADES rents the Underworld from him on favorable terms.
As Greek Mythology developed, TARTARUS descended even further — into the oblivion of obscurity. Now he is little more than a shadow, while his name is a byword for the dark places of punishment for those judged guilty of unspeakablenesses.