Short answers, pulled from the story.
Tartarus is both a primordial deity and a place in the ancient Greek cosmos. In Hesiod's Theogony, Tartarus ranked as the third being to exist, after Chaos and Gaia. As a location, it lay below the realm of Hades itself, described by Zeus in the Iliad as "as far beneath Hades as heaven is above earth."
The gods of Olympus banished Cronus and many fellow Titans to Tartarus after defeating them in war. Cronus himself had earlier imprisoned the one-eyed Cyclopes and the hundred-armed Hecatonchires there. Mortal sinners such as Sisyphus, Tantalus, and Ixion were also condemned there for offenses against the gods.
Sisyphus was condemned to roll a boulder up a mountain slope for eternity; each time it nearly reached the top, it rolled back to the bottom. The punishment was Zeus's response to Sisyphus presuming his cleverness surpassed that of the gods.
In Plato's Gorgias, three judges evaluated the dead: Rhadamanthus for Asian souls, Aeacus for European souls, and Minos as the deciding vote for Greek souls. Souls deemed incurable were condemned to Tartarus permanently as a warning to the living, while those with curable crimes could be purified there. In the Republic, the Myth of Er reported that punishments lasted ten times the span of each earthly crime.
The noun Tartarus does not appear in the New Testament, but the verb tartaroō, meaning "throw to Tartarus," appears in 2 Peter 2:4, describing fallen angels committed to chains in darkness. The English Standard Version gives the Greek reading as a footnote. Adam Clarke argued that Peter's use of Titan-linked language showed the Greeks had preserved a memory of the punishment of fallen angels.
In Book VI of the Aeneid, Virgil described Tartarus as expansive, enclosed within three perimeter walls, beyond which ran a flaming river called the Phlegethon. A hydra with fifty black, gaping jaws guarded the screeching gate. Inside stood a wide-walled castle with a tall iron turret where Tisiphone, the Erinye of vengeance, kept sleepless watch.