Just to add to what has already been said, you make a background of any shade white simply by lighting it to the level where it is about 1/2 a stop brighter than the subject.
If it starts off as a white background, then it doesn't need a lot of light to make it white, but the darker it is, the more it needs, obviously.
The problem with turning a black background white is the sheer amount of power needed to do it, it's well beyond the capability of any flash head, it typically needs flash generators (pack and head systems, in Americanese) and most people don't have them.
Getting a white background to photograph black just requires that NO light reaches the background. This is easy enough to do with enough space, but very difficult to do in a small space.
I think that a lot of people struggle to understand the concept of the Inverse Square Law, which basically states that if you move the light source twice as far away, you lose 3/4 of the power. The reason for this problem is that our primary light source is the sun, the effect of sunlight follows the Inverse Square Law, but because the sun is 93 million miles away, it doesn't make a scrap of difference whether that light is travelling 93 million miles to someone's head and 93 million miles and 5 feet to someone's feet, the amount will be almost equal on both parts of the subject. But when the light source is from a flash head just 2 feet away from one part of the subject and 20 feet away from another part, the difference in light levels is massive. We use this principle to darken backgrounds (and other things).
So, as a rule of thumb:
White (paper) background if you want the background pure white, but you need to put extra light on it as well.
Black background (not paper or vinyl) if you want the background pure black, avoid getting light on it
Grey for anything else, because you can easily make it quite a bit lighter or darker, and can easily change the colour with lighting gels.