Event Horizons
The event horizon is the point outside the black hole where the gravitational attraction becomes so strong that the escape velocity (the velocity at which an object would have to go to escape the gravitational field) equals the speed of light. Since according to the relativity theory no object can exceed the speed of light, that means that nothing, not even light, could escape the black hole once it is inside this distance from the center of the black hole. A more fundamental way of viewing this is that in a black hole the gravitational field is so intense that it bends space and time around itself so that inside the event horizon there are literally no paths in space and time that lead to the outside of the black hole: No matter what direction you went, you would find that your path led back to the center of the black hole, where the singularity is found.
Black Holes and the Speed of Light
Black holes almost certainly exist, and one of their basic properties is that they trap light. However, it is also true that nothing exceeds the speed of light. In fact, the theoretical prediction of black holes is due to the General Theory of Relativity, which is built on the principle that the speed of light in a vacuum is constant. The analogy of a cannonball falling back to Earth with the trapping of light in a black hole is only a crude and suggestive one that is not correct at a fundamental level (for one thing, the cannonball has mass, but light does not; it turns out that this difference is critical, because massless particles MUST travel at light velocity, but massive particles CANNOT travel at light velocity).
To understand fully why a black hole can trap light but the light still always travels at constant velocity requires an understanding of the General Theory of Relativity, but the essential point is that the black hole curves spacetime back on itself, so that all paths in the interior of the black hole lead back to the singularity at the center, no matter which direction you go (an analogy in two dimensions is that no matter which direction you go on the surface of the Earth in a "straight line" (what mathematicians call a "geodesic" or a "great circle"), you never escape the Earth but instead return to the same point. Imagine extending that analogy to the 4 dimensions of spacetime and you have a rough explanation for why light travels at light speed, but cannot escape the interior of a black hole. Singularities Clothed and Naked
The singularity is the point of infinite density thought to exist at the center of a black hole. We have no way of understanding what would happen in the vicinity of a singularity, since in essence nature divides our equations by zero at such a point, and you probably learned some time in math class that you cannot divide by zero and get sensible mathematics. There is an hypothesis, called the "Law of Cosmic Censorship" that all singularities in the Universe are contained inside event horizons and therefore are in principle not observable (because no information about the singularity can make it past the event horizon to the outside world). However, this is an hypothesis, not rigorously proven, so it is conceivable that so-called "Naked Singularities" might exist, not clothed by an event horizon. If such were the case, we can only guess at this point what that would imply for physics near such an object.
