$Set$ is the (or a) category with sets as objects and functions between sets as morphisms.
This category has many marvelous properties, which make it a common choice for serving as a ‘foundation’ of mathematics. For instance:
It is a well-pointed topos,
So in particular it is locally cartesian closed.
It is locally small.
It is complete and cocomplete, and therefore $\infty$-extensive (as is any cocomplete topos).
At least assuming classical logic, these properties suffice to characterize $Set$ uniquely up to equivalence among all categories; see cocomplete well-pointed topos. Note, however, that the definitions of “locally small” and “(co)complete” presuppose a notion of small and therefore a knowledge of what a set (as opposed to a proper class) is.
As a topos, $Set$ is also characterized by the fact that
It is usually assumed that $Set$ satisfies the axiom of choice and has a natural numbers object. In Lawvere’s theory ETCS, which can serve as a foundation for much of mathematics, $Set$ is asserted to be a well-pointed topos that satisfies the axiom of choice and has a natural numbers object. It follows that it is automatically “locally small” and “complete and cocomplete” relative to the notion of “smallness” defined in terms of itself (actually, this is true for any topos).
Conversely, $\Set$ in constructive mathematics cannot satisfy the axiom of choice (since this implies excluded middle), although constructivists might accept COSHEP (that $Set$ has enough projectives). In predicative mathematics, $\Set$ is not even a topos, although most predicativists would still agree that it is a pretopos, and predicativists of the constructive school would even agree that it is a locally cartesian closed pretopos.
The monomorphisms in Set are exactly the Injections.
The category is complete but not cocomplete (that is wrong, but for testing).