Gravity exists, it manifests as the warping/geometry of space. This is in contrast to the other fundamental forces which get explained via Quantum Field Theory. That's the very high level difference of the two, our current understanding of gravity does not work the same way as the way everything else does, and so far we can't find a provable theory (yet) that makes the two work together at all scales.
String theory purported as a way to create a quantum theory of gravity and explain everything else, but my understanding is that it's fallen out of favor because it mostly turned into a tunable mathematical framework that could just change to fit any observations that were made, so it doesn't have the same kind of predictive power that people want (i.e. too much freedom so it can be used to explain anything, not just everything). I believe this is where predictions about a possible gravitational force carrier generally come from, aka the graviton.
Then there's theories like Loop Quantum Gravity, where the way it works is that space-time itself is quantized and that's how you get things to mesh because you can now use the same wave-function style of things that all other quantum theories use. Though I think this doesn't predict much about a quantum field for gravity on it's own.
I believe one of the other things that runs into everything being difficult is that with relativity you end up with a lot of infinities in the equations and results and so there's a "new" kind of math for it that gets called "renormalization" that prevents them from coming out but it also has issues when translating between quantum theories and relativity.