Black hole entropy Bekenstein and Hawking derived the expression for black hole entropy as,
$$
S_{BH}={c^3 A\over 4 G \hbar}.
$$ 
We know from the hindsight that entropy has statistical interpretation. It is a measure of possible number of microstates that is consistent with a particular macrostate. 
My question is how far string theory predict this result.
 A: In 1996, Strominger and Vafa showed that there are indeed $C\cdot \exp(c^3 A / 4G\hbar)$ microstates ($C$ is a subleading factor not affecting entropy) in a particular extremal black hole with a macroscopic horizon in five-dimensional spacetime.

http://inspirehep.net/record/415163?ln=en
http://arxiv.org/abs/hep-th/9601029

This work has been followed by 1,600+ other papers

http://inspirehep.net/record/415163/citations

which have extended this work to various near-extremal, non-extremal black holes, seven-parameter families of black holes (in various dimensions), and even rotating black holes in an ordinary four-dimensional spacetime (which only borrows some "core" of the string calculation). In all of them, the Bekenstein-Hawking entropy has been confirmed by a calculation that is a priori totally different and independent so all these matches look like small miracles (although they directly follow from string theory's consistency as a theory of quantum gravity).
Also, for certain black holes, infinitely many subleading corrections have been calculated and compared with general relativity (Wald's formula). Grey body factors have been computed microscopically, too. Some correct quantitative behavior of black holes and their microstates has also been verified in AdS/CFT, Matrix theory, and other approaches.
Most typically, the exponentially large number of microstates is counted by Cardy's formula, an expression for the density of high-energy states in a conformal field theory.
Today, there's no reasonable doubt that string theory correctly and microscopically reproduces thermodynamics of all black holes that appear as its classical solutions. It's also the only approach where the correct black hole entropy could have been calculated microscopically. That doesn't mean that everything has been understood about black holes in string theory; the recent controversy about firewalls is an example of that.
A: I just want to add to Lumo's answer: The paper by vafa and Strominger instigated a lot of work in determining the statistical formulation of entropy in black holes. Although it must be pointed out that most of these are for cases with supersymmtry and (near) extremal conditions at small couplings. There has also been work in trying to address the microscopic description of black holes at large couplings and this is the Fuzzball paradigm by Mathur, Lunin et.al. This argument bypasses the firewall and has had a lot of success in reproducing the correct thermal spectrum of Hawking radiation. 
