Visibility is distance shading/fog used in the BUILD Engine. It is originally set to whatever number is passed to gamestartup via DEFAULTVISIBILITY in USER.CON. 0 is considered fully bright (no fog). The higher the value, the more fog.
The player structure member visibility allows you to edit this on the fly through cons. If not constantly set it will put itself back at the value specified in DEFAULTVISIBILITY. This is useful for many effects, including creating a lantern, or flashing the screen when you fire (as with the flash command or EXPLOSION2), but without making the shading/fog fully bright.
This is also used in the sector struct.