See book on diff geom.

Differentiable manifold

Introduction to differentiable manifold – formal definition of differentiable manifold

A differentiable structure is what we need to add to a set in order to do differential Calculus on it. In particular differentiation (derivatives) of functions defined on the set. It is what it's needed to define differentiability of functions in a consistent way. It also allows to do Integration (what do you integrate? Differential forms)

Some common manifolds. The differentiable structure induces a topology (Topology)

Relation between differentiable and topological structures

Some invariants

Riemannian manifold

Topological manifold

Locally homeomorphic to Euclidean space

Differentiable manifolds are automatic topological manifolds, as the required local Diffeomorphism to R^n is a homeomorphism.