Short answers, pulled from the story.
Tantalus is most famous for his punishment in Tartarus, where he stood in a pool of water beneath a fruit tree, unable to eat or drink because the fruit and water always receded from his reach. His name is the origin of the English word "tantalize."
Ancient sources disagree. The Nostoi fragment says he asked Zeus to let him live as the gods do. Diodorus Siculus and Ovid say he shared divine secrets with mortals. Apollodorus names both the theft of ambrosia and the sharing of secrets. The story of feeding his son Pelops to the gods, though best known today, appears comparatively late in surviving texts.
In early Greek sources, the more common punishment was a giant rock suspended above Tantalus's head, which he perpetually fled. The rock variant appears in fragments of Archilochus, Alcman, Alcaeus, and Pherecydes, and is the punishment depicted in the Nostoi. The receding water and fruit, familiar from Homer's Odyssey, became the dominant image later.
Tantalus was the father of Pelops, who became the father of Atreus, after whom the House of Atreus was named. The house was marked by persistent misfortune and became the subject of many Greek tragedies, including Seneca's Thyestes, in which the ghost of Tantalus is summoned to sow enmity between Atreus and his brother Thyestes.
Tantalus was sometimes called King of Phrygia, but ancient sources generally placed his city at the western edge of Anatolia, in territory associated with Lydia. The geographer Strabo said his wealth came from the mines of Phrygia and Mount Sipylus. Pausanias recorded a port and a sepulcher bearing his name in that region, and Lake Karagol near modern Izmir has been called Lake Tantalus since antiquity.
In his first Olympian Ode, written around 518-438 BCE, Pindar explicitly rejected the cannibal feast story, calling it a lie and stating it was better to speak well of the gods. He offered an alternative account in which the gods dined with Tantalus without incident, and Poseidon carried off Pelops out of desire, with envious neighbors inventing the rumor that Pelops had been eaten.