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 $K$.

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

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.