Short answers, pulled from the story.
The Theaetetus is a philosophical dialogue by Plato, written in the early-middle 4th century BCE, that investigates the nature of knowledge. Socrates and the mathematician Theaetetus examine three definitions of knowledge: as perception, as true judgment, and as true judgment with an account. All three are found unsatisfactory, and the dialogue ends in aporia.
The main characters are Socrates, the young Athenian mathematician Theaetetus, and Theodorus of Cyrene, a mathematician who explored incommensurable quantities. The dialogue is framed by Euclid of Megara, founder of the Megarian school of philosophy, and his friend Terpsion, who listen as a slave reads the recorded conversation aloud.
The Gettier problem was introduced by Edmund Gettier in his 1963 paper "Is Justified True Belief Knowledge," which drew directly on the Theaetetus and the Meno. Gettier demonstrated scenarios in which a justified true belief is not sufficient for knowledge, challenging the definition that the Theaetetus had examined centuries earlier.
The dialogue is set on the same day as the Euthyphro, immediately before Socrates walks to the Porch of the King Archon to face a hearing for impiety. The trial that hearing leads to ends in Socrates's execution in 399 BCE. The frame narrative, in which Euclid and Terpsion read the dialogue, takes place after a battle at Corinth in which Theaetetus received fatal wounds.
Wittgenstein sent a copy of the Theaetetus to his friend Maurice O'Connor Drury, writing that Plato in this dialogue is occupied with the same problems that he himself was writing about. He later quoted it in Philosophical Investigations. He also criticized the dialogue to O.K. Bouwsma, complaining that Theaetetus shows no fight and calling Socrates's interlocutors weaklings.
The Theaetetus is considered one of the founding works of epistemology because it is among the earliest surviving philosophical texts to systematically examine the nature of knowledge, propose and test formal definitions, and demonstrate why each definition fails. Its influence runs from the Skeptical Academy of the 3rd century BCE through Neoplatonism of the 6th century CE and into 20th century analytic philosophy via Edmund Gettier's 1963 paper.