Short answers, pulled from the story.
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. It must demonstrate that the statement is true in all possible cases, not merely in many examples, which separates it from empirical or inductive reasoning that only establishes reasonable expectation.
The word proof derives from the Latin probare, meaning to test. Related words include English probe, probation, and probability, along with Spanish probar, Italian provare, and German probieren, all carrying the sense of testing or trying.
Euclid, working around 300 BCE, introduced the axiomatic method still in use today. It begins with undefined terms and axioms assumed to be self-evidently true, then proves theorems from that basis using deductive logic. His work Elements was read by educated people in the West until the middle of the 20th century.
The main methods include direct proof, proof by mathematical induction, proof by contraposition, proof by contradiction, proof by construction, proof by exhaustion, combinatorial proof, nonconstructive proof, and probabilistic proof. Each establishes a conclusion in a different way, from combining earlier theorems to constructing an explicit example.
The first proof of the four color theorem was a proof by exhaustion with 1,936 cases. It was controversial because the majority of those cases were checked by a computer program rather than by hand, raising concern that a program error or run-time error could call its validity into question.
Q.E.D. stands for the Latin quod erat demonstrandum, meaning that which was to be demonstrated, and it marks the end of a proof. A common alternative is a small square or rectangle called a tombstone or Halmos, after Paul Halmos, and Unicode provides the end of proof character U+220E.