NDVI is the acronym for normalized difference vegetation index. It is a simple formula using two satellite channels. If one band is in the visible region (VIS, for example AVHRR band 1) and one is in the near infrared (NIR, for example AVHRR band 2), then the NDVI is (NIR – VIS)/(NIR + VIS).
The reason NDVI is related to vegetation is that healthy vegetation reflects very well in the near infrared part of the spectrum. Green leaves have a reflectance of 20 percent or less in the 0.5 to 0.7 micron range (green to red) and about 60 percent in the 0.7 to 1.3 micron range (near infra-red). The visible channel gives you some degree of atmospheric correction. The value is then normalized to the range -1<=NDVI<=1 to partially account for differences in illumination and surface slope.
How is it useful?
NDVI provides a crude estimate of vegetation health and a means of monitoring changes in vegetation over time. The possible range of values is between -1 and 1, but the typical range is between about -0.1 (NIR less than VIS for a not very green area) to 0.6 (for a very green area).