A Singularity is a point in space where matter is packed infinitely densely.

It is where a large amount of matter is packed into a single point in space, and so it has infinte density.
(Density = Mass / Volume, where Volume = 0)

This causes some weird things to happen around it. The denser an object is, the more the gravitational force at its surface. For example, if the earth were to shrink so that its diameter halved, the surface would experience a 4-fold increase in gravitational strength.

The equation for calculating surface gravity is the following:

G is the gravitational constant.
M is the mass of the body.
r is the radius of the body (distance from center to surface (assuming it is spherical (which most astronomical bodies are)))

So if the radius of a singularity is 0, then gravity would be infinite at the surface.

The speed of light (c), the fastest speed at which information can travel through the universe, is finite (300,000 km/s). Now, if we keep compressing a mass closer and closer to 0 volume, at some point we will reach a point where escape velocity exceeds the speed of light.
This means there is a sphere where the escape velocity is the speed of light. It has a special name: the event horizon.