What does the term "background compression" mean? How would you tell you're achieving it, and how would you achieve it? Ideally, the right answer would also explain how this is related to background blurring. For example, is compression completely different, or does it sometimes give you some blurring.
"Background compression" is part of how we perceive perspective in a photograph. Images taken with a narrow field of view (longer focal lengths) will appear to have a shorter back-to-front distance than those with a wide field of view (shorter focal lengths).
It's important to remember that perspective, technically speaking, does not depend on the lens, only on your distance to the subject. If you take a shot with a wide-angle lens and crop down to the very centre, you would have the same background compression as if you took the shot with a telephoto lens (though likely not the same image quality, thanks to the crop and enlargement).
The perception of perspective in a photo is related to the fact that even though they capture wildly different fields of view, we view most photos at approximately the same size (in print, on screen, etc). A wide-angle shot is crammed into a smaller angle of view, and the narrow field of view of a telephoto shot is expanded. If you printed a wide-angle shot large enough (and stood close enough), it would appear perfectly natural.
It follows from this that you're always achieving background compression simply by standing where you stand; it's a much tricker question to figure out how to use it to your advantage compositionally. One example of using a lack of compression (background expansion) would be using an ultra-wide to convey a sense of wide-open space.
This isn't directly related to background blurring; it's possible to have a photo sharp through the entire frame exhibit various degrees of compression. Characteristics of the lens (aperture, focal length, etc) will determine the amount and qualities of background blur.