Using a black (fabric etc) backdrop is helpful, but really, you could have a white wall as your background, and if you can control the light properly, you could photograph it as black.
The key is to ensure that the background is darker than your subject, and obviously set your exposure for the subject, not the background. You will want the background to be at least two stops lower/darker than your subject.
With most studio type photography, you will probably want to light part/most of the front of your subject (so they show up in the photo). But this likely means that the light you have pointing at your subject, is also pointing at the background. So the trick here is to have your subject (and thus the light) farther away from the background.
Most things are characterized with at least some diffuse reflection, and when this is the case, you can use the inverse square law to help you out. The law states that the brightness of the reflected light is inversely proportional to the distance from the source. In other words, if your subject is 2 feet from the light and the background is 4 feet, the background will reflect 1/4 as much light. (2 times the distance, 2 squared is 4). If the subject is 2 feet and the background is 6 feet, the difference is 9 times (3 squared is 9).
So by placing your light close to your subject, but far from the background, you should be able to get a dark/black background.
Another key is to constrain the light, keeping it off the background. This is hard with umbrellas because they bounce the light all over. A softbox, on the other hand, directs all it's light in one direction. You can also use something to 'go between' (GoBo) the light and the background.