Glossary
Crop factor
Crop factor is the ratio between a full-frame sensor's diagonal and a smaller sensor's, and multiplying a lens's focal length by it gives the field of view that lens will show on the smaller sensor.
Also called focal length multiplier, equivalent focal length
How it works
APS-C is roughly 1.5x, Canon APS-C is 1.6x, and Micro Four Thirds is 2x. A 58mm lens frames like an 87mm on APS-C and like a 116mm on Micro Four Thirds.
The lens does not change. Its focal length, its aperture, and the image it projects are identical. The smaller sensor simply records a smaller part of that image, so the result looks like a longer lens pointed at the same scene.
Depth of field at a given framing does change, because you stand farther back to frame the same subject on a smaller sensor, and distance is what governs depth of field. This is why the same lens gives less background blur on a cropped body at matched framing.
Why it matters when buying
Adapting to APS-C or Micro Four Thirds turns standard vintage fifties into portrait lenses and makes vintage wides hard to find. If you want a wide field of view on a cropped body, start below 35mm.