Non-reduced schemes have the very interesting geometric property that they effectively equip points with "infinitesimal orientations" similar to directional derivatives in differential geometry. It's worth actually drawing some non-reduced affine curves and seeing how they behave by looking at the algebra. The motivation for the definition of formal smoothness, for instance, is based on this observation, namely that given a map $X_{red}\to Y$ of $S$-schemes, with $X$ an affine scheme, we want to be able to lift this map to a map $X\to Y$. That is, formal smoothness of $Y$ says that $Y$ is locally nice enough to accomodate infinitesimal deformations of maps into it from affines.
I'm not sure that this was the original motivation for allowing nilpotents, but it's clear that nilpotents give us a much richer picture of the geometry.
Edit: To avoid any confusion, please note that the thickenings of the form $X_{red}\to X$ will not always work if $X=Spec(A)$ does not satisfy good enough finiteness properties (in particular, the nilradical should be a nilpotent ideal, which can fail spectacularly away from Noetherian rings). In general, the requirement is that we have, for any square-zero nilpotent thickening of affine schemes over $S$, $Spec(T/J)\to Spec(T)$ (where $J^2=0$ is a nilpotent ideal of $T$) and any map of $S$-schemes $f:Spec(T/J)\to Y$, there exists a map of $S$-schemes $\tilde{f}:Spec(T)\to Y$ extending the map $f$.
