What is black hole entropy?
Two important thermodynamic quantities are temperature and entropy. Temperature we all know from our fevers, weather reports and ovens. Entropy however is foreign to everyday life for most people.
 
Suppose we have a box filled with gas of some type of molecule called M. The temperature of that gas in that box tells us the average kinetic energy of those vibrating molecules of gas. Each molecule as a quantum particle has quantized energy states, and if we understand the quantum theory of those molecules, theorists can count up the available quantum microstates of those molecules and get some number. The entropy is the logarithm of that number.

When it was discovered that black holes can decay by quantum processes, it was also discovered that black holes seem to have the thermodynamic properties of temperature and entropy. The temperature of the black hole is inversely proportional to its mass, so the black hole gets hotter and hotter as it decays.

The entropy of a black hole is one fourth of the area of the event horizon, so the entropy gets smaller and smaller as the black hole decays and the event horizon area becomes smaller and smaller.

But until string theory there was not a clear relation between quantum microstates of a quantum theory and this supposed black hole entropy.

Black holes and branes in string theory
A black hole is an object that is described by a spacetime geometry that is a solution to the Einstein equation. In string theory at large distance scales, solutions to the Einstein equation are only modified by very small corrections. But it has been discovered through string duality relations that spacetime geometry is not a fundamental concept in string theory, and at small distance scales or when the forces are very strong, there is an alternate description of the same physical system that appears to be very different.

A special type of black hole that is very important in string theory is called a BPS black hole. A BPS black hole has both charge (electric and/or magnetic) and mass, and the mass and the charges satisfy an equality that leads to unbroken supersymmetry in the spacetime near the black hole. This supersymmetry is very important because it results in the disappearance of messy quantum corrections, so that precise answers about the physics near the black hole horizon can be found by simple calculations.

In the previous section we learned that string theories contain objects called p-branes and D-branes. Since a point can be thought of as a zero-brane, a natural generalization of a black hole is a black p-brane. And there are also BPS black p-branes.

But there’s also a relationship between black p-branes and D-branes. At large values of the charge, spacetime geometry is a good description of of a black p-brane system. But when the charge is small, the system can be described by a bunch of weakly interacting D-branes.
In this weakly coupled D-brane limit, with the BPS condition satisfied, it is possible to calculate the number of available quantum states. This answer depends on the charges of the D-branes in the system.

When we go back to the geometrical limit of the equivalent black hole of p-brane system with the same charges and masses, we find that the entropy of the D-brane system matches the entropy as calculated from the black hole or p-brane event horizon area.

This was a fantastic result for string theory. But can we now say that D-branes provide the fundamental quantum microstates of a black hole that underlie black hole thermodynamics? The D-brane calculation is only easily performed for the supersymmetric BPS black objects. Most black holes in the Universe probably have very little if any electric or magnetic charge, and are very far from being BPS objects. It’s still a challenge to compute the black hole entropy for such an object using D-branes.