In GR, gravity is an emergent property of mass interaction with the geometry of 4D spacetime. The more mass, the more spacetime is curved by the massive object.

In QM, gravity is a force, and efforts to express gravity in a quantum framework use the graviton, a massless force carrier for gravitation.

We don't necessarily expect this to exist, the graviton is simply a mathematical tool if you don't like the idea. In relativity, it makes no intuitive sense to have a force carrier for gravity, as gravity is not a force in the same way the others are, it's a property of the geometry of spacetime.

In GR, spacetime is continuous, there is no minimum or maximum length. In QM, spacetime is discrete. There is a minimum quantity.

So already, you can see some incompatibility, because both GR and QM make many predictions that have proven both models to be pretty good descriptions of nature. GR has been more or less proven by experiments with time dilation, gravitational lensing, and more recently gravitational waves. QM has been supported by many experiments too, double slit experiment, experiments with quantum entanglement, you name it, QM has predicted it and with extreme accuracy.

So you have two descriptions of reality that work on two different scales, which is sort of acceptable. Use the two at the appropriate scale and you have the Standard Model of physics, and that mostly works. Except when you have to tackle something like a black hole or the big bang where the scale is quantum level, but the mass-energy is relativistic.

So we want a quantum theory of gravity, and the infinities emerge when trying to formulate a quantum theory of gravity. Specifically, gravitation is non-renormalisable, i.e. at high enough energies, our usual methods to deal with divergences toward infinity doesn't work and itself results in problems with infinities.

Physicists have been working for the best part of a century to unify the fundamental forces, well what does this mean? It means that if you put enough energy into a system, like the amount you'd have at the big bang, it is possible to demonstrate that the electromagnetic force and the weak nuclear force are one force, and "grand unified theories" refer to theories in which the strong nuclear force and gravitational force are similarly found to be all part of one force at the unification energy.

Unification is far from complete, but we have at least a decent picture of the electromagnetic, weak and strong forces individually, in quantum mechanics.

The problem with unification is primarily gravitation, which is extremely weak compared to the other fundamental forces, and we don't know how to describe it in quantum mechanics without using a representative particle. And even then, as you can tell, we haven't gotten all that far.

Most efforts have produced purely mathematical extensions to the Standard Model, like String Theory. Problem with string theories is that they do not make predictions that we can check, and thus the theories are not falsifiable. And there could probably be n different string theories that all work and stitch GR and QM together, but no way to determine which string theory us right.

A lot of people dislike the idea that string theories and some other alternative quantum gravity theories propose which is that there are more than 4 dimensions, anywhere from 5 to 26 to n dimensions.

The reason why people don't like that is that we have no idea what these extra dimensions would look like, we wouldn't even see them, they're basically just more co-ordinates but it doesn't seem very satisfactory just to say there's extra dimensions and that gravity leaks out that way.