This is how I've always understood 'gravity' in terms of relativity, hope this helps:
The presence of mass warps the grid of space (rather, space-time), such that it is simply easier for matter to travel in the direction of other matter. Since matter is always moving around rapidly in many different directions at a subatomic level (unless it is at a temperature of absolute zero), it's movements in the direction of other matter travel greater distances with the same amount of energy, due to space-time being warped; the observed effect being the phenomenon of gravity.
Now, this is not always stronger than the elecromagnetic force. The EM force of a single atom is stronger than that atom's gravitational force, but the collective gravitational force of many atoms (for example, the atoms making up the mass of a star) can overpower the EM force of a single atom on the subatomic level. When this occurs, nuclear fusion happens; gravity pushes the atomic nucleus of an atom past the repulsive electromagnetic field of another atomic nucleus, where it is within range of that nucleus's strong nuclear force (the force holding together the quarks of that nucleus's proton(s)), binding the nuclei together into a single nucleus - a heavier element. It's worth noting that electromagnetism is also contributing to fusion, and not just gravity; inside a star where fusion is taking place, the EM repulsion of other atoms is also what's pushing atoms into each other to cause fusion.
I've always thought of the energy released from nuclear fusion as being all the stored photons of one (or perhaps both) of the atomic nuclei being 'given up' as the entropy of its momentum has to change to the same level of entropy as that of the other nucleus.
The reason this "momentum" is stored up can be explained by relativity -- all motion is relative; so even objects which appear to be moving very slowly relative to each other could actually be moving at much higher speeds relative to another frame of reference. That "invisible momentum" is what's being released in nuclear fusion.
The "big problem" between quantum physics and relativity is that measuring the "warp in the grid of spacetime" really only becomes possible on a macroscopic scale (the scale of celestial bodies such as planets and stars), so it becomes impossible to incorporate gravity into equations regarding interactions at the subatomic level. Since gravity is the result of the curvature of space itself and isn't mediated by a boson (like the other fundamental forces are), it doesn't fit into our subatomic physics models (which explain phenomenon in terms of interactions between subatomic particles -- no such interaction takes place with the phenomenon of gravity).