Explain proof by contradiction. Would anyone mind explaining me how proof by contradiction works? I have a very vague understanding of it so I always avoid using it when it comes to discrete math. From what I've seen its something that is extremely useful when it comes to proofs. 

From what I understand proof by contradiction is the following. Given an example I have to prove it to be true. I take the complete opposite from it and prove it to be false(?), from then on I can assume that the original example is true(?). If anyone would mind correcting my understanding, please do. Also, what would be a good step-by-step guide I can have in my mind when using it?
As Sherlock Holmes said, "When you have eliminated the impossible, whatever remains, however improbable, must be the truth".

If you can prove that not-$P$ is impossible, then you conclude that $P$ must be true.
Proof by contradiction is simple: If $\;\neg A, \Sigma \vdash \bot\;$ then $\;\Sigma \vdash A\;$.


we add the negation of what-is-to-be-shown to our premises
we demonstrate that doing this leads to nonsense (that is, a contradiction).
we conclude the premises entail what-is-to-be-shown.