Hades is the god of the underworld and a son of Cronos and Rhea. He has a part-time wife, Persephone (see this page for her story); it seems they had one daughter. Hades' sole constant companion is Cerberus, a three-headed dog. His realm, also called Hades, is said to consist of three parts, which each compare to a part of the afterlife in Dante's Divine Comedy (which was inspired by a combination of classical literature and Christianity).
The deepest region of Hades is Tartarus, which compares to Hell. Unlike the stereotypical Christian Hell, it's reserved for only the worst of sinners. Sisyphus, a man who literally cheated death twice, rolls a huge boulder up a hill. At the top, it falls and he has to start over. Tantalus, who killed his children and served them as dinner, stands chin-deep in a pool of water, and a low branch on a fruit tree hangs above him. When he tries to drink the pool of water, it recedes, and when he tries to pick a fruit, the bough raises itself up out of reach. Ixion, who tried to marry Hera, turns on a wheel of fire for all eternity. The middle part is Asphodel, which equates to Purgatory. It's meant for the people who are "in the middle", so to speak. It's an endless field of blue asphodel flowers, where billions of souls wander. It's... okay, but boring. The highest section is Elysium, or Paradise; the best of poets and musicians, and mortals who had children with gods, go there after death.