What do we need to define a category? I used some results from the category theory without thinking about its foundations. However, after reading a few topics on MSE, this subject haunts me.
My question is:

What do we need to define a category?

According to some books, a category consists of a class $\text{Obj}$ of objects and a set $\text{Hom}$ of morphisms which satisfy some axioms. For me it means, that to define a category we need some set theory. But there are many different set theories. Do they raise different category theories?
Also, as I understand, when we are talking about specific categories, like $\text{Set}$, $\text{Grp}$,... we mean models (interpretations) of the axioms of a category. Is it correct? 

First, one needs to adopt a foundation of mathematics to define a set, category, and other mathematical objects!
Different foundations give different category theories. Some 'categories', which are called big in one foundation, do not exists in another, e.g. the functor category between two large categories and the localisation of a category with respect to a proper class (large set) of its morphisms. The meanings of the terms (small) set and (proper) class, and the operations you can perform on them, depend on the adopted foundation. Shulman's Set theory for category theory and Mac Lane's One universe as a foundation for category theory discuss the effect of the foundation on the resulting category theory, although the latter is more focused on the advantages of a specific foundation.
