Proof is the evidence that shows something is true or valid. When you show the logical steps that take you from your hypothesis that the world is round to the conclusion that it is, you're formulating a proof.
Photographic prints that haven't been touched up are also called proofs, and that’s also what you call the first copy of a printed text, before it's checked for errors and corrected. When you check a text for errors, you're proofing. If your poker strategy is foolproof, even somebody who knows nothing about poker — a poker fool, that is — can’t get it wrong. Proof is a word you’ll see on bottles of alcohol, too: 90-proof rum means it's 45% alcohol.