Its a technique used by Nikkor lenses to stabilize the gear while shooting. Other companies use different name, however basics are the same eg. canon uses Image Stabilization (IS), optical stabilization by Sigma etc. Its useful when shooting at low shutter speeds particularly at longer focal length or when one has shaking hands or if one is shooting from moving platform like vehicle. It can be also helpful in taking video.
For any image to be sharp, the shutter speed (1/s) must be more than or equal to operating focal length. Ideally the ratio should be 2. When shooting speed is lower, one can still get sharper images without compromise in quality (wrt noise) with increased ISO.
I have shot a video showcasing the effect of VR. I used Nikon D7000 with Nikkor 105 macro IF-ED VR lens for the purpose, a prominent accessory for macro photography . Ofcourse its hand-held, VR - It doesent works when mounted on tripod anyways (unless its VRII). Initially it starts with VR OFF, at 00.10 seconds VR is turned ON. After switching VR ON, see how the jerks and shakes from the video are all gone, showing smooth streaming.
Here are some links to the articles explaining the VR technique:
1. From Nikon
2. General info from Wiki
3. interesting graphical interpretation