Categorical description of algebraic structures There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a statement of Cayley's Theorem that every group G is isomorphic to a subset of the symmetric group on G (see the aside at the bottom of this post... I'm still a little confused on this).
Assuming that I will make this clear in my own mind in the future, are there similar categorical descriptions of other algebraic object, eg rings, fields, modules, vector spaces? If so, what does the Yoneda Lemma tell us about the representability (or otherwise) of those objects?
In particular, are there `nice' characterisations of other algebraic objects which correspond to the characterisation of a group arising from Cayley's Theorem as "subgroups of Sym(X) for some X"?

Aside to (attempt to) work through the details of this: If $C$ is a category with one object $G$, then $h^G=\mathrm{Hom}(G,-)$ corresponds to the regular action of $G$ on itself (it takes $G$ to itself and takes the group element $f$ to the homomorphism $h_f(g)=f\circ g$). Any functor $F:C\to\mathbf{Set}$ with $F(G)=X$ gives a concrete model for the group, and the fact that natural transformations from $h^G$ to $F$ are 1-1 with elements of $X$ tells us that $G$ is isomorphic to a subgroup of $\mathrm{Sym}(X)$... somehow?

Let $C$ be a category, $C'$ the opposite category, and $S$ the category of sets.
Recall that the natural embedding $e$ of $C$ in the category $S^{C'}$ of functors from $C'$ to $S$ is given by the following formulas. 
$\bullet$ If $c$ is an object of $C$, then $e(c)$ is the functor $C(\bullet,c)$ which attaches to each object $d$ of $C$ the set $C(d,c)$ of $C$-morphisms from $d$ to $c$.
$\bullet$ If $x:c_1\to c_2$ is in $C(c_1,c_2)$, then $e(x)$ is the map from $C(c_2,d)$ to $C(c_1,d)$ defined by 
$$
e(x)(y)=yx.
$$
In particular, if $C$ has exactly one object $c$, then $e$ is the Cayley isomorphism of the monoid $M:=C(c,c)$ onto the monoid opposite to the monoid of endomorphisms of $M$.
One can also view $e$ as an anti-isomorphism of $M$ onto its monoid of endomorphisms.
