These definition are as given in a book. I need some clarification of doubts on this topic.
Definition : Let $ \mathcal{A} = {(x_{\alpha},U_{\alpha})}_{\alpha \in A}$ be an atlas on a topological manifold M. Whenever the overlap $U_\alpha \cap U_\beta$ between two chart domains is nonempty we have the change of coordinates map $ x_\beta \circ x_\alpha^{-1} : x_\alpha(U_\alpha \cap U_\beta) \to x_\beta(U_\alpha \cap U_\beta)$. If all such change of coordinates maps are $C^r$-diffeomorphisms then we call the atlas a $C^r$-atlas.
Definition: A maximal $C^r$-atlas for a manifold $M$ is called a $C^r$-differentiable structure. The manifold $M$ together with this structure is called a $C^r$-differentiable manifold.
In the first definition what is meant by change of co-ordinates map ? What is the intuition behind this ? How is it exactly done. Why is the change of co-ordinate map needs to be a $C^r$ diffeomorphism ? Diffeomorphism is a mapping from one differentiable manifold to another with some properties. How can one use the term 'diffeomorphism' while defining a differentiable manifold ?