I'm inspired by Yuhao's question. The functor that takes a scheme S to the set of k-dimensional vector subbundles of C^n x S (understanding "subbundle" to mean that the quotient by it is another vector bundle) is represented by the Grassmannian G(k,n). What functor is represented by the Schubert subvarieties of G(k,n)?

The Schubert variety associated to a sequence of numbers d = (d_0,d_1,...,d_n) is the collection of k-planes that meet the standard i-plane in a subspace of dimension $\geq$ d_i. (The d for which the associated Schubert variety is nonempty are naturally parameterized by k-element subsets of {1,2,...,n}, or more usefully by partitions that fit in a k x (n-k) box.)

It's tempting to say that the functor represented by a Schubert variety should take a scheme S to the set of subbundles of C^n x S that, fiber-by-fiber, meet the standard i-plane in a subspace of dimension $\geq$ d_i. But I don't know what this should mean on a non-reduced scheme.

One way to look for a moduli interpretation for the functor represented by a variety X is to find one or several "universal" or "tautological" families of things over X. Then one hopes that families of things of the same type over another scheme S will be pulled back along maps from S to X. The natural-looking tautological objects over a Schubert variety are the families of intersections $V \cap \mathbf{C}^i$, where V runs through k-planes belonging to the Schubert variety. These families are not equidimensional, and in particular not flat; this might be an obstacle.