For the various types of manifolds, see Dieudonne: Panorama. The definition of manifold includes the existence of charts and transition functions. The definition of PL manifold, differentiable mfd, real analytic and complex analytic mfd involves conditions on the transition functions.
Given a (differentiable???) mfd M, have a notion of G-structure on M. Specific choices of G leads to the notions of Riemannian structure (orthogonal gp), pseudo-Riemannian str (Lorentz group), symplectic str (symplectic grp), and almost complex str (complex general linear group). Any complex analytic mfd comes with an almost complex str, but the converse is true only if the almost complex str is “integrable” (I think).
http://www.ncatlab.org/nlab/show/G-structure
http://nlab.mathforge.org/nlab/show/manifold
There are various things by Thurston available on giga.
http://mathoverflow.net/questions/116814/torsion-in-cohomology-of-smooth-manifolds
Manifolds: Many references in the Manifolds folder.
nLab page on Manifold