There are several ways to deal with this and define a proper class.
A proper class is a large discrete category. But since a large category is usually defined as having a proper class of objects, this just moves the bubble under the wallpaper to ‘large category’ and something must be applied there.
A proper class is a class that is not a set. So now we have to define ‘class’.
A proper class is a class whose cardinality is not the cardinal number of any set. This is a less evil version of the previous definition; however, in some foundations these are actually equivalent (using the axiom of replacement).
A class is a collection of sets. Here the bubble is moved to ‘collection’, but we will be able to pop that bubble below. Also we might want to allow the members of a proper class to be other than sets (such as structured sets); certainly it is true, however, that a pure class is a collection of pure sets.
A class is a formula in the language of set theory for a truth value, equipped with a specified free variable for a set. This is a formalisation of the previous definition, but it must be interpreted metamathematically: a formula for a class in a given context is a formula for a truth value in the extension of by one more free variable for a set.
A class may even be an undefined concept; the real definition is to define a set as a class that is itself a member of some class. With appropriate axioms, this is equivalent to the previous definition (and conservative over set theory without classes), but it's also possible to apply stronger axioms here; this choice is the difference between and as extensions of ZFC.
A class is a subset of a Grothendieck universe , while a (small) set is merely an element of . This gives a relative notion, depending on . As stated here, we get a concept of class like that of the strong theory ; to be more like (and therefore conservative over set theory without an axiom of universes) we should define a class to be a subset of that is definable in the language of set theory.
Category theorists care about proper classes because many examples of categories in practice (such as Set, to begin with!) are large.