What is the incompatibility between Quantum Mechanics and General Relativity?

If you want full details on this subject, 'The Elegant Universe' By Brian Greene is a fantastic book that discusses this topic

Simply, the incompatibility between the two theories is that the use in extremes.
Firstly, relativity generally deals with the massive, and quantum mechanics with the macroscopic. However, when these two are combined, the equations cannot be used together (i don't really know the maths of it, but i think you get stupid answers like infinity). An example of this is in the first few moments after the big bang, and in black holes, where there are very massive objects packed into macroscopic areas.
Secondly, quantum mechanics predicts that on scales of less than planck length (i.e. unimaginably small) the fabric of spacetime would not be flat, but covered in bumps and ripples and trenches etc. (think of fractals, if you know what they are). This would go against many of the assumptions of general relativity.

To understand why string theory solves this problem, you first need to know, that things can only be observed at small scales by thing that are small - our eyes use photons (light particles) but these are too big at these small scales for the imperfections to show up. However, according to quantum mechanics, you could use increasingly small particles with increasingly high energy to detect the imperfections. This works because particles are zero-dimentional and point-like. With strings, however, they are one-dimentional, and if they are the smallest constituents of the universe, then you cannot observe any imperfections smaller than they are, so if they do exist, they would have no effect on anything else.

This is a simplified explanation, and i am not sure of all the details, but i hope it helps.

Does string theory solve the problem?  The answer depends on what you think the problem is.  If the problem is to resolve the conflict between QM and GR, then no, string theory doesn't do that.  The problem that string theory is supposed to address, is to find one consistent explanation for all of the known fundamental forces.  String theory attempts to be that explanation, rendering both GR and QM obsolete in the process.

The incompatibility is in the math, and it's in the basic philosophy.  Issac Newton believed that gravity was a force "acting at a distance."  Modern scientists don't like to believe in action at a distance.  In QM, forces that appear to act at a distance, actually are caused by things continually exchanging invisible, "virtual" particles.  For a long time, believers in QM have sought evidence of a "graviton" particle that would "carry" the gravitational force in the same way that photons are supposed to carry the electro-weak force and "gluons" carry the strong nuclear force.  Einstein took a whole other approach.  He said, that gravity is not really a force at all, but rather an illusion that is due to the inertia of bodies following trajectories (a.k.a., "geodesics") in curved spacetime.

The curvature of spacetime lies at the heart of the math incompatibility.  A whole new branch of mathematics called differential geometry was... not exactly invented by Einstein, but fleshed out by Einstein in order to formulate GR.  QM on the other hand is formulated in flat, Euclidean spacetime, and (I'm going to confess here, I don't really understand the math) attempts to make it work in a curved spacetime have met with failure.

String theory, in its many variants, attempts to replace both of 'em, but based on what I've heard, it's still not ready for prime-time.