The bare facts
Athenian, roughly 470 to 399 BCE. Son of a stonemason and a midwife. Fought as a hoplite in at least three campaigns and by several accounts was conspicuously brave. Married to Xanthippe, three sons. Poor, barefoot, physically unattractive by Athenian standards, and unbothered by any of it.
He spent his days in the agora asking people what they meant by the words they used. In 399 BCE Athens tried him for impiety and corrupting the young, convicted him by a modest majority, and executed him with hemlock.
Three sources, three Socrateses
This is the Socratic problem, and it's genuinely unresolved.
Plato gives us the philosopher: relentless questioner in the early dialogues, then increasingly the vehicle for Plato's own metaphysics in the later ones. Most of what people mean by Socrates comes from here, and the later portion is probably not him.
Xenophon gives us a practical, conventional, rather dull man dispensing sensible advice about household management. Some scholars think this is closer to the truth and that Plato made him interesting.
Aristophanes, in The Clouds (423 BCE), gives us a ridiculous figure in a hanging basket teaching students to argue their way out of debts. It's a comedy, written while Socrates was alive and in the audience, and Plato thought it helped get him killed.
None of the three is disinterested. The honest position is that we know what Socrates meant to the people who wrote about him.
The method, precisely
The elenchus. He asks someone confident to define a term: what is justice, what is courage, what is piety. They answer. He asks a series of further questions, each accepted individually, that together contradict the definition. The interlocutor is left knowing less than when they started.
Crucially, he almost never supplies the answer. The early dialogues typically end in aporia, impasse. The point isn't to arrive at a definition; it's to demonstrate that the confidence was unearned.
His famous claim, from the Apology, is that the oracle called him the wisest man in Greece and he concluded this could only mean he alone knew that he knew nothing.
Why Athens killed him
The charges were impiety and corrupting the youth. The context matters more than the charges.
Athens had just lost the Peloponnesian War, endured the brutal rule of the Thirty Tyrants, and restored its democracy. Two of Socrates' close associates, Alcibiades and Critias, were prominent in that catastrophe. He was a visible critic of democracy in a city that had nearly lost it, and he'd spent thirty years humiliating prominent men in public.
He could have proposed exile and probably got it. Instead he proposed, as his penalty, that Athens should feed him at public expense for life. The jury voted for death by a wider margin than it convicted him.
Reading him
Plato's trial dialogues are where to start: Euthyphro, Apology, Crito and Phaedo, about 150 pages in one volume. Our Socrates quotes guide covers the famous lines, including the ones he never said.
365 Days of Philosophy places him at the start of the Western sequence, one page a morning.
Common questions
Why did Socrates not write anything?
He seems to have thought writing was inferior to live conversation, because a text can't respond to questions or adjust to its reader. Plato records that objection in the Phaedrus, which is itself a written text.
How do we know about Socrates?
Through Plato, Xenophon and Aristophanes, who describe three quite different men. This is called the Socratic problem, and it's unresolved.
What is the Socratic method?
Asking someone to define a term, then asking further questions they accept individually which together contradict the definition. It usually ends in impasse rather than an answer, which is the point.
Why was Socrates executed?
Officially for impiety and corrupting the young. In practice he was a visible critic of a democracy that had just nearly collapsed, associated with men who had betrayed it, and he antagonised the jury at his own trial.
Did Socrates say 'I know that I know nothing'?
Not in that form. In the Apology he reasons that the oracle called him wisest because, unlike others, he doesn't believe he knows what he doesn't.