Before I begin, note that mathematics is an exercise in thought. Mathematics says nothing on what carbon atoms on another sheet of carbon atoms represent - merely how certain ideas like numbers and functions correlate.
Strictly speaking, what mathematicians mean by an actual proof is a formal proof. To quote wikipedia:
A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language) each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference.
For example, when working with integers, it can be an axiom that $1\in\mathbb N$, and another axiom can be that if $a,b\in\mathbb N$ then $a+b\in\mathbb N$. From this we can deduce that $1+1\in\mathbb N$.
The mathematical field of logic has a lot of tools to notate these sequences formally, and also offers various rules of inference (most notably modus ponens), but ultimately no piece of physical paper can be - ahem - proven to be a proof.
However, hardly any real-life proofs are written in this formal system, because it's quite a hassle. For real analysis, this was done in the famous work Principia mathematica, but the formal proof that $1+1=2$ takes two days, a graduate course in set theory and the sacrifice of a goat to understand.
What mathematicians practically mean by "proof" is then anything which gives a clear hint as to how to write a formal proof. This means that the "usual steps" of simplifications and expansions are abbreviated as one step or skipped altogether. Indeed, there is no fine line between a proof and a proof sketch, because every proof is effectively just a sketch, strictly speaking.
There is no way to distinguish a proof sketch from an invalid argument. Sometimes the best thing you can do is convince the author of the proof sketch that he is incorrect, and mathematicians do that by asking about the details of a proof. "Why can you make this simplification?" "Does this always hold?" Sometimes you know that the theory you're working in, such as real analysis, does not have inconsistencies (ie. everything that is provable is actually true), and in that case you can show a proof is incorrect by giving counterexamples. This is not always as easy as it sounds.