The idea of a bundle chart and bundle atlas. The definition of a bundle charts and bundle atlas is rather obscure in my opinion. Is it fair to say that:
1) the purpose of a bundle chart is to give coordinates to each tangent space?
2) the purpose of a bundle atlas is just as an atlas on a manifold to give a correspondence between the coordinates that each  chart in $(1)$ induces?
 A: I'll assume you're talking about vector bundles, for concreteness. Let $E \to M$ be a rank $k$ vector bundle over a smooth manifold $M$ of dimension $n$. Then a bundle chart for $E$ is a pair $(U, \phi)$, where $U$ is an open subset of $M$ and $\phi\colon \pi^{-1}[U] \to U \times \Bbb R^k$, where for each $x \in U$, the restriction $\phi|_{E_x} \colon E_x \to \{x\}\times \Bbb R^k$ is a linear isomorphism. The operations in $\{x\} \times \Bbb R^k$ are induced from the $\Bbb R^k$ factor. A bundle chart effectively takes all the fibers over points in $M$ and straighten them out in a "product fashion" $U\times \Bbb R^k$. The issue is that globally, the fibers $E_x$ (for $x \in U$) inside $\pi^{-1}[U]$ might be not quite straight (or "parallel" to each other, and here I use the word "parallel" in a very loose sense). 
So local trivializations take fibers over an open set, and "organize" them. In general, one cannot do this globally.
Another way to see this is: since the $E_x$'s are vector spaces, they have bases. But can you choose the "same" basis working for all the $E_x$'s, at least for $x$'s ranging over a small open subset of $M$? Yes, and that's precisely what the bundle chart does, by defining $e^\phi_i(x) = \phi^{-1}(x,e_i)$, where $(e_i)$ is the standard basis of $\Bbb R^k$. We may call $(e_i^\phi)$ the local frame for $E$ over $U$ associated to $\phi$.
If you want to compare this with manifold charts, note that $E$ naturally carries a manifold structure, as follows: if $(U,\varphi)$ is a manifold chart for $M$ and $(U,\phi)$ is a bundle chart for $E$ (we assume the same domain $U$ for both, reducing it if necessary), we define a manifold chart for $E$ by the composition $$\pi^{-1}[U] \stackrel{\phi}{\longrightarrow} U\times \Bbb R^k \xrightarrow{\varphi\times {\rm Id}_{\Bbb R^k}} \varphi[U]\times \Bbb R^k \subseteq \Bbb R^{n+k}.$$
In terms of trivializations and local frames, what is happening? If $\psi \in E$, then $\psi = \psi_x \in E_x$ for some $x \in M$. The bundle chart looks at the linear combination $\psi_x = \sum \psi^i e^\phi_i(x)$ and maps it to the pair $(x, (\psi^1,\ldots, \psi^k))$, where the first entry $x$ is a point in the base manifold. The manifold chart, in turn, writes $\psi_x = \sum \psi^i e_i^{\phi}(x)$, and maps it to the $(n+k)$-uple $(\varphi^1(x),\ldots, \varphi^n(x), \psi^1,\ldots, \psi^k)$, to be seen as coordinates of $\psi$.
That being said, we can summarize the discussion by trying to give more or less direct answers to your questions:
1) "the purpose of a bundle chart is to give coordinates to each tangent space": of course we're generalizing tangent spaces to fibers here, but this is morally correct if one thinks of "coordinates" as in "coordinates of a vector with respect to a basis in a vector space", not as "coordinates in a manifold". Treating $E$ as a manifold, the coordinates are given by the above composition.
2) "the purpose of a bundle atlas is just as an atlas on a manifold to give a correspondence between the coordinates that each chart in (1) induces": in the same way that an atlas in a manifold is a collection of charts overlapping smoothly and covering a manifold, you can say that a bundle atlas in a vector bundle is a collection of local frames for the bundle for which the matrices of transition (between different frames at each point) depend smoothly on the base point in $M$. So, if by "coordinates" here you mean "coordinates of a vector with respect to a basis in a vector space" like in (1), then yes, your understanding of a bundle atlas is correct.
