Physicist: Quantum Mechanics (QM) and relativity are both 100% accurate, so far as we have been able to measure (and our measurements are really, really good). The incompatibility shows up when both QM effects and relativistic effects are large enough to be detected and then disagree. This condition is strictly theoretical today, but in the next few years our observations of Sagittarius A*, and at CERN should bring the problems between QM and relativity into sharp focus.
Relativity comes in two flavors: special and general. Special relativity describes how time and distance are affected by movement (especially fast movement), and it replaces Newtonian mechanics, which is only accurate at low speeds. Einstein came up with it by looking at the mathematical repercussions of the fact that all of physics works the same way, independent of movement (constant speed is the same as no speed). Special relativity has been exhaustively tested (relativistic effects have been verified all the way down to walking speed), and works so perfectly that it is now held up as the yardstick against which all new theories are tested. In fact, QM would make grossly inaccurate predictions if Dirac hadn’t shown up and tied QM together with special relativity to create “relativistic QM”.
General relativity, on the other hand, describes the stretching and bending of space and time by gravity. Einstein came up with it when he thought about what the universe would be like if inertial and gravitational acceleration were the same (turns out they are). By the way: gravitational acceleration is what pushes you toward the ground, and inertial acceleration is what pushes you back into the car seat when you step on the gas. It’s general relativity that causes the problems. Here’s two (of a possible untold many):
1) Smooth vs. Chunky: General relativity needs space to be “smooth”, or at the very least continuous. So if you have two points side by side, then no matter how close you bring them together you can still tell which one is on the right or left. Quantum mechanically you have to deal with position uncertainty. At very small scales you can’t tell which is right or left. In addition (as the name implies) QM requires everything to be “quantized”, or show up in discrete pieces. You see this clearly with atoms, photons, and even phonons (which is quantized sound! How awesome is that!?). Less clear is the quantization of space, which would require space to be “chopped up”. This choppiness will never be directly measured. The predicted “chunky scale” should be no large than 10-35 m. For comparison, a hydrogen atom is about a million, million, million, million times larger (10-24).
2) The Information Paradox: According to general relativity when stuff falls into a blackhole everything about it’s existence (with the exception of mass, charge, and momentum) is completely erased. That doesn’t sound so bad. We tend to think of blackholes as being like galactic garbage disposals. However, if all the information about something is destroyed, then you lose time-reversibility. Time-reversal is the idea that if you run time backwards, all the basic physical laws of the universe continue to work the same. More obscurely, you can predict the future based on what you know now, and time reversal means that you can derive what happened in the past as well. QM requires that time-reversibility (or “unitarity”, to a professional) holds. So QM requires that blackholes cannot destroy information. One way around this is amazingly complicated entanglement between all of the in-falling matter, and all of the Hawking Radiation that comes out later. Again, we’ll never be able to measure this. To get results we would have to exactly measure at least half of all of the photons generated by Hawking radiation over the essentially infinite life time of the blackhole (every blackhole that exists today will be around long, long after the heat death of the universe).