Let $M$ be a topological space.
Let $d$ be a natural number.
Let $k \ge 1$ be a natural number.
A $d$-dimensional differentiable structure of class $\CC^k$ on $M$ is a non-empty equivalence class of the set of $d$-dimensional $\CC^k$-atlases on $M$ under the equivalence relation of compatibility.
Also defined as
A $d$-dimensional differentiable structure of class $\CC^k$ is sometimes defined as a maximal $C^k$-atlas of dimension $d$. See Bijection between Maximal Atlases and Differentiable Structures.