How exactly does string theory claim to unify quantum theory and gravity into one theory?

This is a far, far longer question than someone can just answer in a couple minutes.  String theory postulates extra dimensions (and postulates that particles in the regular 4 dimensions are manifestations of strings in the many dimensions).  It then explains why gravity is weaker--because it spills out in the extra dimensions whereas the other forces are confined to the regular old 4-D bulk/membrane we know and love.  The wiki page is a good starting place if you want to read more.

String theory is controversial because it has failed to produce any meaningful results (beyond explaining the relative weakness of gravity, which was the reason it was cooked up in the first place).  It makes few predictions, and the ones it makes generally fail.  Feynman quote: "String theorists don't make predictions, they make excuses."  It sucked up the attention of a large portion of the smartest theoretical physicists for 30 years, so if there were something to it, we'd expect better.

Bekki is correct as usual.  But I'd read "The Elegant Universe" by Brian Greene if you are more than just casually interested.  Greene is very easy to read and he has excellent examples or illustrations.  "Universe" describes the history of getting to string/M theory, including why they started working on such a theory in the first place.  And, to be fair, Greene also lists many of string/M theory's failings, some of which Bekki mentioned.  By the way, the string/M theory community as an official web site [See source.]  You should read it over to get the string stuff straight from the horses mouth...so to speak.

To your points:

What is so revolutionary about string theory: the ideas.

Higher (10, 11 or more) dimensions.  Infinitely thin vibrating strings of Plank's length (~10^-33 cm).  The vibrating frequencies determine how each string will appear in our 4D space (e.g., mass or massless, photon, electron, graviton).  Gravity in our universe is weaker than it ought to be; so perhaps some of our gravity exists outside our universe.

-How string theory claims to unify quantum physics with gravity-- macro world.

The mathematics of string theory actually predicts the graviton, the highly theorectical messenger particle that tugs at mass (i.e., the force of gravity).  And the graviton is a quantum, like the photon.

-And why is there a recent controversy sorounding string theory? Why has it 'failed' in certain respects.

String/M theory has failed to be a "theory" because we don't yet have the ability to test it.  Much of Einstein's way out thinking has been tested and found to be true.  

None of string theory has been tested; so we don't know if it is true or not.  The large Hadron Collider [See source.] currently under construction near Cern may be able to do some tests of the theory.  Just one verification of one part of the theory would be remarkable and help elevate string/M theory to more than just a WAG based on fancy math.

VERY simplistically, string theory removes a central conflict between relativity and quantum mechanics.  Quantum mechanics is NOT consistent with "singularities" or "point-particles"--basically any object of finite mass that takes up zero volume.  Einstein's Theory of General Relativity, however, does not rule out their existence, and in fact indirectly predicts the existence of black holes, which are objects with finite mass and zero-volume.

When any attempt is made to combine theories, the mathematics always falls apart because of this--you get some equation which ends up being zero divided by zero, which is unworkable.

The usefulness of string theory lies in that instead of thinking of matter as point-particles, they can be thought of as objects that take up space, and as such, there is no point at which you need to divide zero by zero.

Unfortunately, string theory is almost TOO good for this purpose.  Not only does it give you an answer, it gives you AT LEAST 6 different answers, all of which could be right.  Much of modern string theory work is trying to prove that all 6 of these answers are actually the same, because if they are ACTUALLY different, equivalent answers, than the theory loses all usefulness.

This is SUPER simplistic, and for a better lay explanation, try Briane Greene's "The Elegant Universe" and "Beyond Superstrings"