One way to think of categories algebraically is as a "monoid-oid". So, do for monoids what you do for groups to make groupoids. A (small) category is a set, with an associative, unital, partial multiplication, i.e. composition. So, anything that is an example of this, (monoids, groups, abelian groups, groupoids, etc) is certainly representable as a category.

But there are other examples of things which we aren't used to thinking of as partial monoids. Pre-orders for instance can be thought of as a set of arrows, where the identities are the set elements in a the usual sense and multiplication "witnesses" transitivity. This seems to be kindof a "Lawvere-esque" way of thinking about the category (cf. elementary theory of abstract categories, ETAC).

Warning: this could be a digression... A better way to recover lots of concrete things in a categorical way is to look at functors that preserve some structure (e.g. products or monoidal structure) as the "things" and natural transformations as the "thing homomorphisms". Lawvere theories give a good example of this, and provide a general way to talk about groups, rings, etc. Incidentally, they do pretty much the same job as monads (and there are some nice equivalences). PROPs are another good example. These are both specific cases of a type of 2-monad called a Doctrine. For a prolonged, but interesting discussion, see here.