* To an analyst, compute the colimit along $K\subseteq\mathbb R^n$ of the spaces $C^\infty_K(\mathbb R^n)$ of smooth functions on $\mathbb R^n$ with support contained in $K$. You can build othr spaces which will ring a bell in this way, like $L^1_{\mathrm{loc}}$, and friends.

* For an algebrist, construct the localization of a commutative ring as a colimit.

* For the complex analyst, construct the space of germs of analitic functions as a colimit.

* For a more general audience, convince them that directed limits and colimits are just unions and intersections of sorts. For example, show them that a ring/module is the colimit of its finitely generated subobjects, that the cantor set is a colimit, etc.

* Do the kernel and the cokernel, of course. Generalize to equalizers and coequalizers to show that you get useful notions in more general categories which serve similar purposes.