G-structure

[Kobayashi 1972]
Definition
For a manifold $M$ the frame bundle $FM$ is a $GL(n)$-principal bundle. Given a subgroup $G$ of $GL(n)$ we call a $G$-structure on $M$ to a $G$-principal bundle $P$ which is a subbundle of $FM$.$◼$

This is a particular case of reduction of group structure with respect to the inclusion $G\to GL(n)$.
Indeed, according to Sternberg, Lectures on differential geometry:
Definition
A $G$-structure on an $n$-dimensional manifold $M$ is defined as a reduction of $FM$ to the group $G$. $◼$

Equivalence to existence of a section

When $M$ and $G$ are given, and $G$ is a closed subgroup of $GL(n)$ the existence problem becomes the problem of finding a cross section in a certain bundle. Since $GL(n)$ acts on $FM$ on the right, $G$ also acts on $FM$. If $G$ is closed the quotient $FM/G$ is also a bundle, in particular is the associated bundle to $FM$ with fibre $GL(n)/G$.
In @KobayashiNomizu1996 can be found:
Proposition 5.6. The structure group $G$ of $P(M,G)$ is reducible to a closed subgroup $H$ if and only if the associated bundle $E(M,G/H,G,P)$ admits a cross-section $\alpha :M\to E=P/H$.$◼$
Remark The correspondence is 1:1.$◼$

See the example below.

Relation to tensors

If we have a tensor field defined in a manifold $M$ it let us to define a subgroup $G$ of $GL(n)$ (the subgroup that leaves invariant the copy of the tensor in ${\mathbb{R}}^n$) and a $G$-structure on $M$.

I suppose the converse is true: a $G$-structure let us define one or several tensor fields in $M$. The group $G$ is possibly (I'm not sure) characterized by some tensors in ${\mathbb{R}}^n$, $T_i$, of type $(r_i,s_i)$, which are their invariants. We can cover $M$ with trivializing neighbourhoods of $E$, $\{U_{\alpha }\}$, and select the G-frames corresponding to $(x,e)\in U_{\alpha }\times G$. Finally, we can define the tensor fields whose expression in this frames is given by $T_i$.

This is better understood in the example below.

Example

The orthonormal frame bundle.
If $G=O(n)$, a $G$-structure on $M$ will be a principal bundle $E$ such that for $p\in M$ the fibre $E_p$ is made of frames such that one can be obtained from other by means of an orthogonal transformation. The choice of the G-structure implies we are pointing out in every $p$ a special kind of basis: the orthonormal ones.

On the other hand, $FM/G$ consists of families of frames that can be transformed into others in the family by means of an orthogonal transformation (orbits of the action). So it is intuitively clear that a cross section $\sigma :M\to FM/G$ (a choice of a family in every $p$) provides us with the same data that $E$ (see orthonormal frame bundle).

At the same time, this is the same as providing the metric tensor. Once we know which frames are considered orthonormal, we can cover the manifold $M$ with trivializing open sets for $E$, $\{U_{\alpha }\}$. Then define the metric $g$ like the one whose local expression in every $U_{\alpha }$ is $g{|}_{U_{\alpha }}\equiv {\delta }_i^j$.

Relation to Cartan geometries

All these ideas are related to Cartan geometry. It seems (I'm not sure enough yet) like if a G-structure $P$ on $M$ together with a connection (a principal connection on $P,$ that is, a $\mathfrak{g}$-valued 1-form with properties) is the same as providing to $M$ a Cartan geometry modelled on $$
G\to \mathbb{R}^n \rtimes G