A black hole is a region of space where the force of gravity is so intense that to escape requires a velocity greater than light.
If you throw a ball up into the air, it quickly returns to the surface of the Earth. The harder you throw it, the faster it will travel when it leaves your hand, and the higher it will go before returning. In theory if you could throw it hard enough it would leave the Earth's surface and go into orbit, or even escape the Earth's grip altogether. The gravity of the Earth will not be enough to return it to the surface if it travels fast enough. The speed that the ball must have in order to escape is known as the escape velocity, which for the Earth is about 11 kilometers per second or 24,000 miles/hour.
As a mass is compressed into a smaller and smaller volume, the gravitational attraction at its surface becomes more severe, as though it is being concentrated. This means the escape velocity at the surface is higher. Objects have to travel faster in order to escape into space. Eventually, if the mass is concentrated enough, a point is reached when not even light travels fast enough to escape. This extreme condition, by definition, is a black hole.
Gravity exerts crushing pressure on a star. Meanwhile, as long as its fuel lasts, the core of a star generates radiation through nuclear fusion. This radiation creates an outward pressure that balances the inward pressure of gravity caused by the star's mass. As the star's fuel is exhausted, gravity begins to win the fight which compresses the star inward. If the mass is high enough the star will eventually collapse. It is believed that high mass stars eventually become black holes.
In the crowded cores of galaxies it is believed that millions of stars have combined to form supermassive black holes. This is the simplest explanation for the high velocities we measure for stars orbiting near galactic cores. Our own galaxy, the Milky Way, is theorized to contain a black hole in the core on the order of 4 million solar masses.