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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.