For a category $C$ it is required that the morphisms of any two objects of $C$ form a set (c.f. Lang: Algebra, or Weibel: An introduction to homological algebra).
What's the point about this requirement ? Would there be any disadvantages / logical deficiencies if one allows the morphisms to form a proper class ?
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, 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 these categories 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.
Edit: Actually if one wants to think of schemes as sheaves on CRing, then you need to think hard about local smallness, otherwise the category of schemes will not be category under the naive definition that the homs are sets.
I have to admit I had never heard of the distinction that Thierry Zell and Adam Hughes point out in their answers; I had always learned that a "small" category as opposed to a "large" one was merely the stipulation that the objects form a set, but that hom-sets were always sets.
That said, in hindsight the reason this is correct is Yoneda's Lemma: that there exists a fully faithful embedding of any category in its functor category to Sets. Of course, if hom-sets are classes then one cannot do this, since there is no such thing as the class of all classes but there is a class of all sets. Since Yoneda's Lemma is arguably the most important single fact about category theory, it is worth preserving.
If you look at Steve Awodey's book Category Theory (Oxford Logic Guides * 49) you'll see that on p. 22, Definition 1.12 is that a category in which $\hom_\mathbf{C}(A,B)$ is a set for every pair of objects $A$ and $B$ is called locally small.
As David mentioned, category theorists generally tend to shy away from such `size issues', because in a sense they do not touch the heart of the matter, as the original question rightfully suggests. Apart from such foundational issues, I can think of two practical reasons where it is important that homs be sets.
First, there is Freyd's celebrated Adjoint functor theorem. It gives conditions that characterize precisely when a given functor has an adjoint. Crucially, one of the conditions, called the `solution set condition', is that a certain class is in fact a set. This shows that size issues do play a fundamental role in category theory, which came as quite a surprise to most people.
Second, one can think of Enriched categories. It turns out that a lot of category theory goes through if homs are not necessarily objects of the category Set of sets and functions, but objects in an arbitrary monoidal category, with composition being a morphism of that category. For example, relating to Kevin and Qiaochu's comments above, a 2-category can be seen as a category enriched in Cat, the category of categories. But this also gives some surprising examples. For example, a metric space can be seen as a category enriched in $\mathbb{R}^+$, i.e. the poset $[0,\infty]$ with monoidal structure given by addition. And of course a locally small category is just a Set-enriched category. This is not an argument against `large categories' per se, but does indicate that a lot of murky waters can be avoided by only considering locally small categories.
