Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a technical theorem as a glorious wellspring of intuition- something, at least from my perspective, that rings are missing; and I want to know why.

Certainly the axiom system is more complicated- so there is no way you're going to get as simple a characterisation as you do with groups- but surely there must be some sort of universal object for rings of a given cardinality, analogous to the symmetric group in group theory. I would be surprised if it was a ring- the multiplicative and additive properties of a ring could be changed (somewhat) independently of one another- but perhaps a fibration of automorphisms over a group? If so is there a natural(ish) way of interpreting it?

Perhaps it's possible for a certain subclass of rings, perhaps it's possible but useless, perhaps it's impossible for specific reasons, in which case: the more specific the better. 

Edit: So Jack's answer seems to have covered it (and quickly!): endomorphisms of abelian groups is nice! But can we do better? Is there a chance that 'abelian' can be unwound to the extent we can make this about sets again- or is that too much to hope for?

Every (associative, unital) ring is a subring of the endomorphism ring of its underlying additive group.  Rings act on abelian groups; groups act on sets.  The universal action on an abelian group is its endomorphism ring; the universal action on a set is the symmetric group.  Modules are rings that remember their action on an abelian group; permutation groups are groups that remember their action on a set.

A set is determined by its cardinality, but for abelian groups cardinality is not a very useful invariant.  Rather than "order" of a ring, consider the isomorphism class of its underlying additive group.  This is even commonly done in the finite ring case, where the order still has some mild control, but not as much as the isomorphism type of the additive group.
