Gram-Schmidt process is a way to convert a set of non-orthogonal vectors into a set of orthogonal vectors, while preserving the subspaces spanned by initial vectors. Gram-Schmidt process in Euclidean spaces is captured in a computational algorithm called QR-decomposition.
QR-decomposition is used in many computational algorithms as an intermediate step, most notably, in calculating eigenvalues of matrices (the QR-algorithm).
In the context of statistics, Gram-Schmidt process and QR-algorithm have many applications:
The attached documents contain an exposition of Gram-Schmidt process, QR-decomposition and a statistical application to Studentized means.