Standard Model in the context of General Relativity General Relativity is a classical theory that states that the matter affects the geometry of spacetime, and in turn, the spacetime geometry influence the paths of free particles, which follows geodesics. As far as I've heard, this matter can also be "field matter".
On the other hand, currently we have a quite accurate theory for the structure of matter and the other interactions, which is QFT and the Standard Model, more specificaly.
Currently, providing a quantum theory of gravity is something that hasn't yet been fully achieved, let alone unify this quantum theory of gravity with the Standard Model.
But could we take the Standard Model and bring it together with classical General Relativity? I mean, considering matter as perceived by the Standard Model, predict how it affects spacetime and considering the curvature of spacetime, predict how it affects the fields from the Standard Model?
If this is possible, what would be a situation where it would be relevant? I would guess something involving cosmology, black holes or dark matter, but I'm probably totally wrong, as I'm new to all of this.
Also, if this is really possible, does this coupling (GR + Standard Model) has a name?
 A: As a rough approximation one can neglect the backreaction of the quantum matter fields on the gravitational background. This would give the Quantum Field Theory in curved spacetime — a well-established (though far from precisely formulated) field. The most important results of QFT in curved spacetime are arguably the Unruh effect (the vacuum state is perceived as a thermal bath of elementary particles by the accelerating observer) and Hawking effect (black holes radiate elementary particles).
Then, one is tempted to consider the backreaction of the quantum fields on the classical background. To my knowledge, no consistent formulation of "semiclassical" gravity exists (see, for example, the Eppley&Hannah's thought experiment).
We could, however, consider small perturbations of the gravitational background around the classical solution and pass to the perturbation theory. This approach is infamously nonrenormalizable, leading to a theory which breaks down close to the Planck energy scale.
Finally, one can treat the gravitational field quantum-mechanically along with the matter fields, which is the subject of the quantum gravity research.
A: Take the current Big Bang model of the universe.


Diagram of evolution of the (observable part) of the universe from the Big Bang (left) - to the present.

After the plane shown on the left, at 380.000 years, is the realm of the standard model as we know it, and the model of the universe uses statistics and thermodynamics,  in a General Relativity kinematics set up to describe the observable universe. Before that line, the standard model is used extensively towards the left of the plot, until the inflation period, where effective quantization of gravity is applied. 
You can see the use of the standard model in the times before in the link here.

Looking backward, the general idea is that back beyond 1 Planck time (10^-43seconds) we can make no meaningful observations within the framework of classical gravitation.

As you state General Relativity is a classical theory and cannot be quantized in a straight forward way.
String theories accommodate quantization of gravity and can embed the standard model naturally, but they are at the research stage as far as picking a specific  string theory to be stated as standard.
