Short answers, pulled from the story.
The Seven against Thebes were seven champions chosen by Adrastus, king of Argos, to lead an army against the city of Thebes. Their purpose was to restore Polynices, the son of Oedipus, to the Theban throne. The most commonly cited list includes Polynices, Tydeus, Amphiaraus, Capaneus, Parthenopaeus, Hippomedon, and either Adrastus or Eteoclus.
The war began from a dispute between the brothers Polynices and Eteocles over the kingship of Thebes. In the most familiar version, first appearing in Euripides' Phoenician Women, the two had agreed to rule in alternate years, but after Eteocles held the throne for the first year he refused to relinquish it. Polynices then went to Argos, married the daughter of King Adrastus, and secured his father-in-law's support for a military expedition to reclaim the throne.
All seven champions died except Adrastus, who escaped on his divine horse Arion. Capaneus was struck down by Zeus' thunderbolt while scaling the city walls. Tydeus was mortally wounded and forfeited his promised immortality when Athena saw him eating the brains of his attacker. Amphiaraus was swallowed alive by the earth when Zeus split the ground before Periclymenus could spear him. Polynices and Eteocles killed each other in single combat.
According to the myth, the Seven against Thebes stopped at Nemea on their march and encountered Hypsipyle, the nursemaid of the infant Opheltes. While she led the Seven to a spring, the unattended child was killed by a serpent. The Seven held funeral games in the child's honor, and those games became the origin of the Nemean Games, one of the four great Panhellenic athletic festivals.
Amphiaraus had foreseen that the expedition was doomed and that all of the champions except Adrastus would die, so he initially refused to go. However, he had previously sworn an oath to let his wife Eriphyle settle any future disputes between himself and Adrastus. Polynices bribed Eriphyle with the golden necklace of Harmonia, and she ruled that the expedition must proceed and Amphiaraus must join. Bound by his oath, he had no choice.
Creon, the new ruler of Thebes, forbade the burial of the Argive dead. In Euripides' Suppliants, Theseus the king of Athens led an army to Thebes, defeated the Thebans, and recovered the bodies, burying them at Eleusis and at Eleutherae. In Aeschylus' earlier lost play Eleusinians, Theseus achieved the same result through negotiation rather than battle. The geographer Pausanias reported seeing ancient unmarked tombs along a small road in Eleusis named the Seven Against Thebes.