Okay, first of all, we need some definitions here. "Singularity" is actually a mathematical term that means a point where a value is undefined, and can refer to any value. You've actually encountered singularities in everday life. For example, as a ball bounces, it loses energy on each bounce, so the frequency of the bounces increases rapidly, until, just before the ball comes to rest, the frequency actually runs away to infinity, giving you a singularity. Another example would be the geographic north and south poles, where longitude isn't defined.
There are several singularities associated with a black hole, several of which do occur at the event horizon. For example, the gravitational time dilation at the event horizon is so great that to an outside observer, time appears to be frozen there. From our point of view, a black hole actually appears to be frozen in the act of collapsing, and one early name for them was a "frozen star." (Let's all be thankful that term didn't catch on.
) However, what's usually meant when people talk about a black hole's singularity is the gravitational singularity at the center. From the point of view of matter falling in to the hole, all the matter should collapse down to a mathematical point, with no dimensions. The density of the matter and the strength of gravity both run away to infinity at this point. However, this is occuring in such an extreme environment that we can't really be sure what actually happens. As DrRocket said, general relativity may simply not apply there. Even if it does, it's very hard to say what the equations actually mean in physical reality. There are several schools of thought about what's going on, and it's still being actively debated.
An event horizon is a place where an observer can no longer receive any information. You shouldn't think of it as a surface or even a particular place, since its apparent location will change depending on where you're looking at it from. It's very much like a regular horizon - a place that you can't see past. It's actually not possible to reach or pass an event horizon - like someone trying to reach the horizon, an event horizon will appear to recede as an observer falls in to a black hole, and from the point of view of an distant observer, the person entering the black hole will appear more and more time dilated until they just appear to stop before reaching the horizon.
One last interesting thing is that event horizons aren't only associated with black holes. For example, to an observer traveling at a constant acceleration, the universe itself appears to have an event horizon. If you accelerate forever, your speed will approach the speed of light, so there is a point behind you from which it is impossible for light leaving there to ever reach you while you're accelerating. In practical terms, for accelerations that humans are capable of, the univers'es event horizon is so far away you'd need an enormous telescope to tell it's there at all.