1. The finished edge of a piece of woven fabric.
2. (Manufacturing) A strip of surplus material, such as the edge on a sheet of postage stamps.
'Selvage' is the U.S. English spelling, 'selvedge' is the British English spelling.
Etymology: Middle English, self + egge (edge)
Selvages prevent the fabric from fraying.
When cutting fabric pieces for quilting, the selvage is usually cut off and discarded: often the pattern is different on the edge (or is replaced by text about the fabric designer, etc.), plus it can shrink and change differently than the rest of the fabric.
But some quilters will save the selvages, to make selvage-only creations.
One quilt I saw at a quilt show was decorated with poetry composed of words found on selvages; I recall the effect was somewhere between refrigerator magnet poetry and a snuggly ransom note.
There are even entire books on sewing with selvages: