I'm sorry, I should have included more explanatory links in what I wrote earlier. When reading what I put together below, please keep in mind that each of these paragraphs is normally an entire grad school course in mathematics. I have attempted to clarify my meaning, but if I haven't succeeded, please note that it took me several years to understand all this. Also, that was 20 years ago, so I no longer recall exactly how I learned it.
$\mathbb{R}^n$ is Euclidean $n$-dimensional space, which is written with $n$ as an exponent, because that space is the product of $n$ copies of the real line with itself. This is different from the raising and lowering of indices used to distinguish between vectors and one-forms, which constitutes much of the machinery of calculation in general relativity.
A "chart" $\phi$ is a function which maps an open subset $U$ of a topological manifold $M$ to $\mathbb{R}^n$. This is a precise statement of what physicists normally mean by "coordinate system", which is an assignment of the values of various measurable quantities (the real number in each of the $n$ dimensions) to each point of $U \subset M$. Homeomorphism means each chart is a continuous function with a continuous inverse, formalizing the idea that $M$ "looks like" $\mathbb{R}^n$. Local means there has to be at least one of these for every point in M, but not every point has to use the same one.
A collection of charts which together cover $M$ (the union of all the $U$ is $M$) constitute an "atlas" as long as they are compatible with each other, which means some condition is satisfied in those places where two charts overlap. That is, if $\phi: U \rightarrow \mathbb{R}^n$ and $\psi: V \rightarrow \mathbb{R}^n$ are two charts in the atlas, then doing first $\phi$ inverse and then $\psi$, or the other way around, is a function from $\mathbb{R}^n$ to $U \cap V$ and back to $\mathbb{R}^n$ again. If that function is diffeomorphic (differentiable with a differentiable inverse) for all pairs of overlapping charts in the atlas, then we call $M$ a differentiable manifold, and the study of such objects (differential geometry) is effectively the most general kind of vector calculus it is possible to do.
If you manage to wade through all of that and still want more, then take a look at what general relativity calls vierbeins (or tetrads), and then try to read about frame bundles. A section of a frame bundle over a manifold is a function which assigns to every point a basis for the tangent vector space at that point, which is again sort of taking what physicists normally mean by "frame" and abstracting it as much as logically possible.
Does that help?
