Many forgetful functors in an algebraic setting are representable; for example, the forgetful functor $\text{Ab} \to \text{Grp}$ is represented by $(\mathbb{Z}, 1)$.  Functors with left adjoints are also representable, and <a href="http://en.wikipedia.org/wiki/Representable_functor#Left_adjoint">sometimes the converse holds</a>.