It's not hard to give a clearer explantaion than Bryan Peterson's book

Good work from CT though
Just some additional comments really. Being guided by an exposure meter reading (which is basically just a best guess) is all very well, and when shooting film that's all we ever had to go by. It was often wrong too, hence the term 'bracketing' - shooting two or three frames at slightly different exposure settings just to make sure. The meter is a measuring device, and a pretty dumb one at that. The user has to add the understanding. In theory, white should be about three stops above 18% grey. There or thereabouts, but it depends on a lot of things such as the sensor's dynamic range, what 'white' is exactly as a tonal value, and the camera/sensor calibration.
Fortunately, digital is different and we have the LCD to be guided by - in particular, the histogram and blinkies (highlight over exposure warning, in the menu). This shows you what you have actually got, with no guesswork involved.
So if the background is grey, forget what the meter might say. It's obviously being fooled and the subject is under exposed - so increase the exposure until it's not. Really, it's as simple as that. But I would also say this, 'white' is never absolutely white. Even the brightest white background will reproduce as a very, very light shade of grey. So you have to push it a bit, maybe adjust the contrast setting in picture styles or tweak it a bit in post processing, if absolutely pure white is necessary. Usually it isn't, but you need to be aware of that. The only other way of ensuring a pure white background is to light it separately from the subject and deliberately over expose it a bit. However, that's obviously difficult if your subject is also sitting directly on the background, which where a professional studio would use a light table where the surface is transluscent and is separately lit from underneath, though that has drawbacks too.
In a nutshell, 100% pure white is hard to achieve, but is usually not necessary, and 95% white should be pretty easy.
The other thing to watch with this sort of work is white balance. Even if it's only slightly out and would normally go unnoticed, with white it will show up. And it's usually important to get it consistent shot to shot. Just do a custom white balance. Sorted. It's really easy - see handbook.