The problem of understanding the local geometry of CR embedded submanifolds in CR manifolds arises naturally in several complex variables analysis, e.g., in the study of isolated singularities of analytic varieties (via their links). Significant work has been done on this in connection with rigidity phenomena and the classification of proper holomorphic mappings between balls. We develop from scratch a CR invariant local theory based on the CR tractor calculus associated to the Chern-Moser-Cartan connection. This produces the tools for constructing local invariants and invariant operators in a way parallel to the classical Gauss-Codazzi-Ricci calculus for Riemannian submanifolds.