Under what conditions does something count as a proof? I'm new to discrete math, and I am struggling with the concept of proof. I asked for a precise definition, but the professor remarked there was no such precise definition.
For example, a homework question asks me to show that, if σ: A→B is an isomorphism for two structures A and B, then σ inverse is an isomorphism. Now, this seems to me to fall directly out of the definition: an isomorphism is a bijection that is closed under operations in (on?) the structure, and that has an inverse. Since σ is a bijection, σ inverse will be a bijection, and so uniquely determined. Since σ is an isomorphism, σ inverse must be an isomorphism. I am sure this does not constitute a proof, but I am unsure why.
In general, I want to know in virtue of what one sequence of observations could be called a proof but not another, and why something that is obvious may yet need to be proven.
 A: There are such things as 'formal proofs', which have a very precisely defined syntax and other organisation, but as a new student to discrete math you will be asked for 'plain' mathematical proofs, and as your professor says, there is no precise definition for that.
Still, the basic idea is that a proof is a step by step demonstration for why something is true. To do that, you start with basic assumptions, definitions, or axioms: statements we hold to be basic truths.  And then we infer other statements from them, one at a time, where we make sure that every inferred statement 'clearly' follows from the statements before it. 
We will insist that every inference is deductively valid' meaning that the inferred statement has to be true assuming the truth of the statement(s) it is inferred from. Indeed, as part of your demonstration, you need to be very clear from which statements you derive the new statements. Moreover, inferences can be deductively valid and yet not be 'obvious', and so you will need to break down such inferences into smaller inferences such that each of those inferences is 'obvious' .
I think you can now see where the imprecision of trying to exactly define a proof comes in: what exactly counts as 'obvious' or 'clear'? Indeed, what may be obvious to some may not be obvious to others. As such, for some audiences a proof demands smaller steps than others. But what exactly is 'small enough'?  This is mostly a matter of experience; your professor is an expert in this, so pay close attention how he/she lays out these proofs, so you get a feeling of the steps and organisation and level of precision that is deemed necessary.
Good luck!
