I am trying to understand the notion of a principal $G$-bundle versus a vector bundle. Here $G$ is a Lie group.
Supposedly, principal $G$-bundles are a generalization of vector bundles. My problem here is that most sources, for example the wikipedia page, talks about bundles over $GL_n(\mathbb{R})$ or some other such matrix group. But the fibers of vector bundles are of the form $\mathbb R^n$ and not $GL_n(\mathbb{R})$. So, how are principal bundles the generalization of vector bundles?
On the other hand, is there some kind of correspondence between vector bundles and $GL_n(\mathbb{R})$-bundles, so that principal bundles are in some indirect way a generalization?
Yes. Given a principal $G$-bundle and a linear representation $\rho : G \to \text{Aut}(V)$, you get an associated vector bundle whose fibers look like $V$ instead of $G$. This gives you a functor from principal $\text{GL}_n(\mathbb{R})$-bundles to $n$-dimensional vector bundles (taking the standard $n$-dimensional representation) which is an equivalence of categories (the inverse functor is given by taking the frame bundle).