If you are doing serious category theory, then at some point you will come across what are affectionately known as 'size considerations' or similar. In particular, any presheaf category $Cat(C,Set)$ and the subcategory of sheaves is not locally small (homs are sets) when $C$ is not a small category (set of objects). For example, you might want to consider the category of sheaves on the category of spaces, or schemes, or on a topos (these are not usually small). Then the Yoneda embedding, as Ryan points out, will not work, which is a bit of a problem.

One workaround is the axiom of [universes](http://www.ncatlab.org/nlab/show/universe), say with two universes $U \in V$. Then you can talk about locally small categories in $U$ - homs are elements of $U$ whereas the objects form a subset of $U$ (so are '$U$-large'). Then the presheaf category consists of functors to the category of sets which are (isomorphic to) elements of $U$. The (pre)sheaf category is then locally small in $V$, and the Yoneda embedding for this category is taken into presheaves with values in the category of sets which are (isomorphic to) elements of $V$.

Whenever you see the phrase 'locally small', you can be sure someone is using some sort of foundations that distinguishes between large and small - Universes, GBN class-set theory or similar - to get around the issue.