The "feather" of a light source is the transition between full light straight on, and no light when a light source is rotated away from the subject. Imagine you have a main light source say at 45 degrees to the line between the camera and the subject and the light source (say a square soft box) is pointed straight at the subject. If you rotate the light source away from the subject (keeping the light at the same location), the light source will appear to become a narrower and narrower rectangle (thinner vertical) until it totally disappears because it it is eventually facing away from the subject.

The great thing about feathering is that, as you rotate and narrow the light say over the first 45 degrees there is little change in the quantity or quality of light, but as you get close to the light disappearing there is a more rapid drop-off. This is the "feather." This is a sweet light to use. Experiment with it by rotating little by little.

An added advantage is when photographing say group of people in a straight line all facing the camera. If the main light is say at 45 degrees to the persons with the face of the source pointing at the center of the group, the end of the line closest to the light will be much brighter than the end furtherest from the light (because of drop-off due to distance). By feathering the light by pointing it at the end of the line of people furtherest away, they see the light as a square, and those closest see the light as a thin vertical light. Consequently those furtherest away get more light (it's full on to them) while those closest get less light (as the area they 'see' is smaller and the quantity of light is dropping off). Thus you get even lighting across the entire line of people.