If the neutron star is massive enough, then the gravitational force can grow strong enough even to overcome the hard-core repulsion between quarks and compress the neutrons themselves, making the gravitational force even stronger until no force can resist the gravitational collapse, at which point the entire mass of the star compresses (theoretically) to a single point called the singularity. We can't tell anything about the singularity for a simple reason: nothing that gets close to it can ever get away again.
The easy, handwaving way to see why is as follows: at any distance from
a massive object, any other object will be in orbit about it providing
it executes circular motion at just the right speed. As you get closer,
the orbital velocity gets higher. Now, for a sufficiently heavy
object, there is some radius at which the nominal orbital velocity is
the speed of light. From inside that radius, called the Schwarzschild
radius (
),
not even light can escape but is inexorably drawn "down" into
the singularity. Thus all light (or anything else!) falling on such an
object's Schwarzschild radius will be perfectly absorbed, which accounts
for the name, " black hole."
We can easily estimate 
using a crude classical approximation: for a masss m in a
circular orbit about a mass M, F=ma
gives 
which
reduces to 
or 
. If 
this becomes 
.
This result is actually off by a factor of 2: the actual Schwarzschild
radius is twice as large as predicted by this dumb derivation:
I have tried to find a simple explanation for this extra factor of two,
but failed. Simply using the "effective mass" 
in place of m makes no difference,
for instance, because it appears on both sides of the equation the same
way. However, I don't feel too bad, because apparently it took Einstein
about seven years to get it right. [The time it took him to develop his
General Theory of Relativity, which explains that extra factor properly.]
A more rigourous description is beyond me, but I can repeat what I've heard and list some of the phenomenology attributed to black holes, of which there are two types: the Schwatzschild (non-rotating) black hole and the Kerr black hole, which spins. Presumably all real black holes are of the latter category, since virtually every star has some angular momentum, but there is probably a criterion for how fast it must spin to qualify as a Kerr black hole.