Let me elaborate on unknown (google)'s comment.  If $\text{Rng}$ denotes the category of unital rings and $\text{Grp}$ denotes the category of groups, then it turns out that "group of units" is a functor $\text{Rng} \to \text{Grp}$.  In fact, it's <a href="http://en.wikipedia.org/wiki/Representable_functor">representable</a>; the group of units of a ring $R$ can be identified with $\text{Hom}(\mathbb{Z}[x, x^{-1}], R)$ (the set of all ring homomorphisms from $\mathbb{Z}[x, x^{-1}]$ to $R$), where the group structure on the Hom-set comes from a Hopf algebra structure on $\mathbb{Z}[x, x^{-1}]$.  And covariant representable functors <a href="http://en.wikipedia.org/wiki/Representable_functor#Preservation_of_limits">preserve limits</a>, including binary products.

Closely related to representability (although I've never been clear on the precise relationship) is the fact that "group of units" has a left adjoint $\text{Grp} \to \text{Rng}$ which constructs the group ring of a group.  And functors which have left adjoints <a href="http://en.wikipedia.org/wiki/Adjoint_functors#Adjoints_preserve_limits">preserve limits</a>.

Presumably many other examples can be handled in a similar vein, although I can't think of any right now, so you'll have to give me more examples.