"Proof" and "prove" both come from the same word (Latin probus). Usually, "proof" is a noun that means "evidence", and "prove" is a verb that means "demonstrate". You prove that an accusation is true by showing proof.
There are a few special cases to be aware of, but they're all technical terms:
- So-much-"proof" as an adjective can refer to the alcohol content of a drink (1 proof is 0.5% alcohol by volume). This comes from an old practice of using gunpowder to provide "proof" in the usual sense of the alcohol content.
- Both "proof" and "prove" can be a verb meaning to let bread rise with yeast.
- "Proof" as a noun can mean an example of a product to be produced, such as a book to be printed or a sample copy of a business card. "Proofreading", meaning to examine a sample copy for errors, comes from this meaning.
- "Something-proof" is an adjective that means "resistant to something", such as a waterproof tent or soundproof walls.
- A mathematical or logical proof is a specific variety of proof that proves that a conclusion is true if certain assumptions are true.
There are a number of other rare uses of "proof", but these are the ones you're likely to encounter. The word proof/prove is a very old one; it at least goes back to Roman Latin, and so it has gained some specific meanings in different fields (such as its use to measure alcohol) that are now far removed from the original processes.