The definition of atlas I encountered is
An atlas for $M$ is a family {$\varphi_\alpha: U_\alpha \rightarrow U_\alpha^\prime: \alpha \in A $} of charts such that {$U_\alpha: \alpha \in A$} is an open cover of $M$.
Intuitively, I guess it is just the index set, a place where holds all the names of the subsets. But this is rather ungrounded.
The entry on wikipedia used $\alpha$ without define which set it belongs to, .
And I checked Clifford Taubes' Differential Geometry, he defined chart without mentioning atlas altogether.
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$\begingroup$You are correct. $A$ is just an index set. It merely gives a label to each chart.
Note that a chart is an element of an atlas; therefore to define a chart you won't mention an atlas. It's the other way around: an atlas is a special collection of charts.
Taubes's book defines an atlas without an indexing set, but the definition he gives is equivalent. This will just change some notation. For example $$\bigcup_{\alpha \in A} U_\alpha$$ would be written $$\bigcup_{(U, \varphi) \in \mathcal{U}} U$$ in Taubes's book.
$\endgroup$0$\begingroup$It is as you've guessed. Where Wikipedia says $\bigcup U_\alpha$, it means the union over all $\alpha \in A$, or just "all alpha" where the universe is implicitly the whole index set, whatever that is. Since the index set isn't otherwise used, it's often considered extraneous in mathematical writing.
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