Given a $k$-vector bundle $p\colon E \to M$ of finite rank $n$, its frame bundle (or bundle of frames in $E \to M$) is the bundle $F E \to M$ over the same base whose fiber over $x \in M$ is the set of all vector space bases of $E_x = p^{-1}(x)$. The frame bundle has a natural action of $GL_n(k)$ given by an ordered change of basis which is free and transitive, i. e., the frame bundle is a principal $GL_n(k)$-bundle.
The frame bundle of a manifold $M$ is the principal bundle $F T M \to M$ (also denoted $F M \to M$) of frames in the tangent bundle $T M$.
In the finite-dimensional case, the dual $GL_n$-principal bundle $(F T)^* M$ is the coframe bundle of the manifold. This means that $F^* M = (F T)^* M$ is the associated bundle to $F T M \times_{GL_n(k)}GL_n(k)$ where the left action of $GL_n(k)$ on $GL_n(k)$ is given by right multiplication by inverses $g. h = h\cdot g^{-1}$. Also $F T M\cong (F T)^* M\times_{GL_n(k)} GL_n(k)$ using the same formula. Furthermore, the right action of $GL_n(k)$ on this associated bundle is given by left multiplication by inverses on $GL_n(k)$ factor.
Coframe bundle $F^* M$ has the following independent description. One looks at the set $\mathcal{U}(M)$ of tuples of the form $(p,(U,h))$ where $p\in U$ and $(U,h)$ is chart of the smooth structure on $M$, $U\subset M$, $h : U\to \mathbf{R}^n$ (an atlas where $U$-s make a basis of topology suffices). $GL_n(k)$ acts on the right on $\mathcal{U}(M)$ by
Then $((p,(U,h))A)A' = (p,(U,h)) (AA')$ holds. The total space $F^* M$ of the coframe bundle by the definition, as a set, consists of classes of equivalence of tuples in $\mathcal{U}(M)$ where $(p,(U,h)) \sim (p',(U',h'))$ iff $p = p'$ and the Jacobian matrix of the transition between charts at $h'(p)$ is the unit matrix: $J_{h'(p)}(h\circ (h')^{-1}) = I$. The left action of $GL_n(k)$ is induced on the quotient. There is an obvious projection $\pi: [(p,(U,h)]\mapsto p$. To define the differential and principal bundle structure one charts $F^* M\to M$ with local trivializations from the neighborhoods of the form $U\times GL_n(k)$, transfers the structure and checks that the transition functions are of the appropriate smoothness class and right $GL_n(k)$-equivariant. The basic prescription is that to every chart $(U,h)$ one defines a map
where $z = [(\pi(z), (U',h'))]$ with $\pi(z)\in U'\cap U$. This does not depend on the choice of the chart $(U',h')$ around $\pi(z)$. There is an equivariance
and on intersection of $(U,h)$ and $(V,g)$
Then $\phi_h$ is onto and
what shows that the transition functions are smooth (where $GL_n(k)$ has the standard differential structure).
Formalization of frame bundles in differential cohesion is discussed there in the section Differential cohesion – Frame bundles
The frame bundle $Fr(X)$ carries a canonical differential 1-form with values in $\mathbb{R}^n$.
This is defined as follows. Let $p \in Fr(X)$ be a point in the frame bundle $\pi \colon Fr(X)\to X$ over some point $x \in X$, hence a linear isomorphism $p \colon T_x \simeq \mathbb{R}^n$. For $v \in T_p Fr(X)$ a tangent vector to the frame bundle, its projection $\pi_\ast v \in T_x X$ is a tangent vector to $X$. Then the value of $\alpha$ on $v$ is the image of this $\pi_\ast(v)$ under the isomorphism $p$
(Sternberg 64, section VII, (2.2))
A choice sub-bundle of a frame bundle which is a $G$-principal bundle for $G\hookrightarrow GL(n)$ defines a G-structure. See there for more.
a section of a frame bundle is also called a frame field.
Wikipedia (English) frame bundle
Shlomo Sternberg, Lectures on differential geometry, Prentice Hall 1964; Russian transl. Mir 1970