There is one particular fact that greatly helps me understand Schubert varieties and Schubert cells, and the serves as kind-of an introduction: They are a generalization of row echelon form for matrices. If $V subseteq F^n$ is a $k$-subspace of the standard $n$-space over a field $F$, then it is well-known that $V$ has a unique basis of row vectors in reduced row echelon form. The shape of the form divides the set of all $V$ (the Grassmannian) into cells which are affine spaces. The closures of such a cell — itself plus lower cells — is a Schubert variety. For example, one pattern of RREF for a 2-plane in $F^4$ looks like this:
$$begin{pmatrix} 1 & * & 0 & * \\ 0 & 0 & 1 & * end{pmatrix}$$
This is clearly an affine cell. It is equally easy to show (in the case of a Grassmannian) that the cells are bijective with the $k$-subsets of an $n$-set.
This is not a complete explanation because a Grassmannian is just the simplest type of flag variety, but it captures the basic geometric idea.
As for references, there is an interesting mini-review for combinatorialists on page 398 of Stanley, Enumerative Combinatorics, Vol 2.
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