General relativity predicts that as an object collapses to form a black hole, it will eventually reach a point of infinite density. What that really means is that the theory of relativity breaks down at this point, and no one knows what happens at the center of a black hole - we would need a viable theory of quantum gravity in order to understand this.
But here's something that you might find useful: when we talk about the "size" of a black hole, we usually talk about something called the Schwarzschild radius. The Schwarzschild radius is the "point of no return" - once you get closer to the black hole than it, you can never escape. Consequently, the escape speed at the Schwarzschild radius is equal to the speed of light, and the value of the Schwarzschild radius works out to be about (3x105 cm) x (M / Msun), where M is the mass of the black hole and Msun is the mass of the Sun. (Typically, M for a black hole in our galaxy is around 10 times the mass of the Sun, but for supermassive black holes at the centers of galaxies it can be millions or even billions.)
There is a rough analogy between a black hole and an atom. In both cases, the mass is concentrated in a tiny region at the center, but the "size" of the object is much bigger. You can use the Schwarzschild radius to calculate the "density" of the black hole - i.e., the mass divided by the volume enclosed within the Schwarzschild radius. This is roughly equal to (1.8x1016 g/cm3) x (Msun / M)2, where M is defined as above. From the point of view of an outside observer, this might as well be the actual black hole density, since the distribution of matter within the Schwarzschild radius has no effect on the outside.
This page was last updated June 27, 2015.