Q:

How does one spot an error in a math proof?

I hope this is not a dumb question but I truly would like to know: How do you know when a proof breaks down and when an error has occurred?

A:

There's no set algorithm for doing so. It would be really really easy to do so if mathematicians always wrote nice proofs following a rigid structure, but that's not really how things work. Mathematicians often cheat and say things like "it's clear that...", followed by some statement that the reader is expected to be able to find themselves without difficulty. In general, regardless of what your purpose with the proof is, reading a proof is going to be an active process, since some assertions will be less obvious than others, or some may be "well, this is tedious, but trust me, it works". If you can follow the proof in detail to the end, then there's probably not an error unless you have some reason to believe that the proof is bad (e.g. if it proves something which is known to be false, it's probably not a good proof).
Often, though, you can identify a premise that is false- like if you see a proof that $1=-1$ and at some point see the assertion that $\sqrt{1}\sqrt{1}=\sqrt{-1}\sqrt{-1}$, you can know there is a problem then, because computing reveals the two sides to be unequal. Often, it is good to substitute known values in for premises, like if you were proving something like
$$\frac{x+y}2\geq \sqrt{xy}$$
you could test it on particular $x$ and $y$. In other situations, you might have a numerical solution and test it against a theoretical one. If you do simple sanity checks like this, you can often backtrack to where a proof went bad. This doesn't tell you "why" a proof is wrong, but it does tell you where it went off the rails, which can allow you to carefully scrutinize the logic leading there and, hopefully, find the flaw. This might mean you have to expand upon the logic presented in a proof - and it could be that a mathematician takes something false as "obviously" true or makes an error in computation, or something which is not immediately obvious and can only be found by careful scrutiny.

