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Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and Richard Dedekind by letter. Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory

A category is some collection of objects and morphisms. Whatever that collection is, it can’t be a set (at least not in the ZFC sense) because of Russell’s paradox, or else it would not be possible to talk about a category whose objects are sets. At a foundational level, that collection is usually defined as a class, which is kind of like one step up from a set: a class contains sets, but a class never contains another class (so you don’t have to worry about the class of all classes). In an alternate terminology, all sets are said to be classes, and a class which is not a set is called a proper class. A category whose collections of objects and morphisms are both sets is called a small category, and if at least one of them is a proper class, it is called a large category. An intermediate level, which comes up quite frequently, is a locally small category. A locally small category is a category where, for every two objects X and Y , the class of morphisms from X to Y is a set. The category S e t , whose objects are sets and whose morphisms are functions, is an important instance of a large category (because its class of objects is not a set, precisely because Russell’s paradox dictates that there is no set of all sets), but which is locally small (for any pre-chosen sets X and Y , you can construct the set of functions from X to Y ; it’s the subset of P ( X × Y ) ). One interesting category is the category of small categories, written C a t , whose objects are small categories and whose morphisms are functors between them. Russell’s paradox is not an issue here, because C a t itself is a large category, so is not one of its own objects. Occasionally referred to is C A T , the category of large categories, but discussing C A T requires a foundation accounting for collections of classes, in which case C A T is not a large category, but a “very-large” category. As such, none of these categories contain themselves as objects. https://www.quora.com/What-is-a-category-in-math-and-why-does-the-category-of-sets-not-violate-Russells-paradox