Usually when you start studying category theory you see the usual definition: a category consists of a class $Ob(\mathcal{C})$ of objects, etc.
If you take ZFC to be your system of axioms, then a "class" (a proper one) is something which you can't formally use, since everything in the universe of discourse is a set.
Some people (MacLane? Grothendieck?) were understandably worried about this. Cutting down on the history which I am unqualified to give an accurate account of, there is the definition of Grothendieck universe.
If we add the following axiom of universes to ZFC, then we can get around having to use classes:
Axiom of universes (U): every set is contained in some universe.
Now, it can be proven that the axiom of universes is equivalent to the
Inaccesible cardinal axiom: for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger.
which was proven to be independent of ZFC. Hence we can work in ZFC+U and do category theory with the concern of dealing with proper classes put at rest.
This seems to be now a standard approach to a good foundation of category theory.
My question, to put it informally, is: how innocent is the axiom of universes?
What I mean by this is: how do we know it does not have unexpected consequences which may alter the rest of mathematics? The motivation was to give a good foundation of category theory, but it would be unreasonable to give a great foundation that altered the rest of ordinary mathematics.
To give an example, we now that adding the axiom of choice to ZF has some startling consequences. For example, the Banach-Tarski paradox. How do we know that ZFC+U does not have some equally startling consequences? Why are we at rest with adding this axiom to our foundation of mathematics? Isn't this a rather delicate question? How much do we know about how good is the universe approach? (I would say a foundation for category theory is better than another one if it solves the 'proper classes issue' and it has less impact on the rest of mathematics.)