A gusset is an extra piece of material that helps expand or strengthen a fabric object such as a pair of pants, a tote bag or a pillow. A gusseted pillow has side panels, rather than just the top and bottom typical of the average bed pillow.
If looking at two different pillows designed for the same purpose -- one with gussets and one without -- the pillow with gussets appears to have side panels or "walls" of fabric stitched between the front and back of a throw pillow or the top and bottom of a bed pillow. Piping might be used as a decorative element around the gussets to make the design stand out even more. Some furniture cushions that look boxy are also gusseted. More fabric is used to make a gusseted pillow than one without gussets that is otherwise the same -- the gusseted version has an added dimension determined by the size of the gusset fabric. Another way to view a gusset is to consider a paper bag made to hold only a greeting card or flat item compared to a paper lunch bag. The paper lunch bag has a gusseted bottom and sides, allowing it to open more fully and stand.