User selecosi gave a beautiful explanation, but perhaps it's a bit too detailed. Trying to give a slightly more high level idea, the conflict is simply that if

User selecosi gave a beautiful explanation, but perhaps it's a bit too detailed. Trying to give a slightly more high level idea, the conflict is simply that if you introduce the gravitational effects (spacetime bending) from general relativity into the equations of quantum mechanics you get infinities all over the place. This happens because in QM particles are not localized, they exist as a probability wave with certain values at any space and time, but the effects of their mass at all those spaces and times adds up to infinity if you just apply the calculations from relativity, in a way that we haven't been able to solve. This is all very roughly speaking - the exact reasons are quite technical and I don't understand them myself. Electromagnetism and other interactions had similar issues, but we found mathematical solutions to fix them, and those solutions we know just don't work with the equations for gravitational interactions.