Principal bundles and fibre bundles Let $\pi_P:P\rightarrow M$ a principal $G$ (right action) bundle. Let $F$ be a manifold with a left action of $G$. Then we have the notion of associated fibre bundle over $M$ whose fibre is $F$. I do not know if there is any connection in the other way which is an inverse for the above construction.
Suppose we start with a $Gl(n,\mathbb{R})$ bundle. We have associated vector bundle whose fibre is $\mathbb{R}^n$ which is a vector bundle.
Suppose we start with a vector bundle of rank $n$, we have the notion of what is called frame bundle which is a principal $Gl(n,\mathbb{R})$ bundle. 
This says that there is a correspondence between vector bundle of rank $n$ and principal $Gl(n,\mathbb{R})$ bundle.

Do we have such correspondence in case of fibre bundles? Given a fibre bundle, $\pi:E\rightarrow M$, with fibre $F$, can I produce a principal $G$ bundle with an action of $G$ on $F$ such that associated fibre bundle is precisely the one I started with? 

 A: Yes. There is a bundle $\mathrm{Fr}(E)\to M$ whose fibre at $m\in M$ is the space $\mathrm{Iso}(F,E_m)$, where $E_m$ is the fibre of $E$ over $m$. Then $G=\mathrm{Aut}(F)$ acts on the right of $\mathrm{Fr}(E)$ by composition. That this is a locally trivial bundle follows from the fact $E$ is locally trivial. Moreover this action makes $\mathrm{Fr}(E)$ a principal $G$-bundle. The canonical action of $G$ on $F$ means we can form the associated bundle $\mathrm{Fr}(E)\times_G F$, and this is (IIRC) isomorphic to $E$.
I didn't specify what kind of isomorphisms should be used. In the case that $E$ is a vector bundle, we take linear isomorphisms, and this is just the usual frame bundle, with $\mathrm{Aut}(\mathbb{R}^n) = GL(n,\mathbb{R})$. For an arbitrary continuous fibre bundle, we take homeomorphisms, and then we have $G=\mathrm{Homeo}(F)$. For a smooth fibre bundle we take diffeomorphisms (but then note that one lands in infinite-dimensional manifold territory, and one has to be a little bit more careful if the fibre $F$ is a non-compact manifold).
