A differentiable manifold of dimension $n$ is a set $M$ and a family of injective mappings $x_\alpha : U_\alpha \subset \mathbb{R}^n \to M$ of open sets $U_\alpha$ of $\mathbb{R}^n$ such that:
1) $\bigcup_\alpha U_\alpha = M$.
2) For any pair $\alpha,\beta$, with $x_\alpha(U_\alpha) \cap x_\beta (U_\beta) = W \neq \emptyset$, the sets $x^{-1}_\alpha(W)$ and $x^{-1}_\beta(W)$ are open sets in $\mathbb{R}^n$ and the mappings $x_\beta^{-1} \circ x_\alpha$ are differentiable.
3) The family $\left\{(U_\alpha,x_\alpha )\right\}$ is maximal relative to condition 1) and 2).
What does maximality mean in this context?
Maximality in this context means that if $\{(V_\beta,y_\beta)\}$ is another collection of injective mappings $y_\beta$ and open sets $V_\beta$ such that $\bigcup_\beta V_\beta = M$, and such that for each $\alpha$ and each $\beta$, the mappings $x_\alpha\circ y_\beta^{-1}$ and $y_\beta\circ x_\alpha^{-1}$ are differentiable where they are defined, then $\{(V_\beta,y_\beta)\}\subset\{(U_\alpha,x_\alpha)\}$.
Another way of stating maximality is to say that the collection $\{(U_\alpha,x_\alpha)\}$ contains every chart $(V,y)$ that is compatible with $(U_\alpha,x_\alpha)$ for each $\alpha$.