A topological manifold $X$ of dimension $n \in \mathbb{N}$ is by definition a topological space that is locally homeomorphic to a Cartesian space $\mathbb{R}^n$. A choice of such morphism
is a coordinate system or coordinate chart or just chart on the image of $\phi$.
This generalises to other sorts of manifolds.
An atlas is the collection of coordinate charts defining a manifold structure.
coordinate transformation?, diffeomorphism