"A black hole is where God divided by zero." - Anonymous
Short version of what a black hole is: A black hole is a massive, dense object whose escape velocity is higher than the speed of light, and that's really all that you need to establish that it is a black hole.
More details. The black hole has what is known as an event horizon, the boundry where the gravity is so intense that not even light can escape. Outside of the event horizon, a black hole's gravity behaves just like the gravity of any other object, like stars or planets. If you replaced the sun with a black hole of the same mass, the planets would still orbit it in exactly the same way. In order to get sucked into a sun-"sized" black hole, an object would have to be on the same sort of trajectory that would cause it to get sucked into the sun. The same goes for black holes of other size - if you had a miniature black hole with the same mass as a comet, only objects that came close enough to be captured by a comet would be captured by the black hole.
The event horizon is located at a distance from the center of the hole called the Schwartzchild radius. This radius is found from the formula for an escape velocity (I could post that here if you're interested). The Schwartzchild radius is the distance where the escape velocity is greater than the speed of light. Consequently, nothing in known physics can get out once it passes the event horizon. If you're inventing newly discovered physics for you book, a ship with a faster than light propulsion system might be able to activate that system from inside the event horizon and escape. This would depend on how the system worked and whether the ship could survive the tidal forces.
Even though it would probably be possible to take a spaceship on a course that would come close to the event horizon without getting sucked in, this still would not be a good idea due to an effect called tidal forces. Gravity follows an inverse square law, diminishing with the square of the radius. A ship flying past the event horizon would experience a much higher level of gravity at the side closest to the black hole than the side furthest from it. The difference in gravity would create very large stresses on the ship's structure, basically stretching it parallel to the gravitational field. These stresses are pretty high and can tear a ship in two. Or the crew members, for that matter. Note that very large black holes would not have such extreme tidal forces until you get inside the event horizon.
There have been some speculations that since the space-time continuum is so heavily distorted around a black hole, the black hole may be an actual hole in the space-time contiuum, leading to another part of space. However, using it as a transportation system would be rather risky. For one thing, it would probably exit at another black hole, so you wouldn't be able to escape the exit. Second, the tidal forces would be so severe that it is unlikely for the ship to survive even approaching the black hole. And to top it off, any theory that the hole might "go somewhere" is pure speculation.
There's a lot more about these that I could post if you're interested.