  We investigate local and metric geometry of weighted Carnot-Carath\'eodory
spaces which are a wide generalization of sub-Riemannian manifolds and arise in
nonlinear control theory, subelliptic equations etc. For such spaces the
intrinsic Carnot-Carath\'eodory metric might not exist, and some other new
effects take place. We describe the local algebraic structure of such a space,
endowed with a certain quasimetric (first introduced by A. Nagel, E.M. Stein
and S. Wainger), and compare local geometries of the initial C-C space and its
tangent cone at some fixed (possibly nonregular) point. The main results of the
present paper are new even for the case of sub-Riemannian manifolds. Moreover,
they yield new proofs of such classical results as the Local approximation
theorem and the Tangent cone theorem, proved for H\"ormander vector fields by
M. Gromov, A.Bellaiche, J.Mitchell etc.
