The waistband is a band of material at the upper hem of trousers, skirts or underwear that strengthens the garment there and allows it to sit tightly at the waist (or somewhat below the waist). In trousers, the waistband is made particularly strong to enforce the garment where the fabric is in greatest risk of tearing, e.g. when sitting down or strongly breathing in.
Belts usually run in loops at the waistband of the trousers.
Waistbands in underwear
Until the end of the 19th century, there were no elastic fabrics, so waistbands were either enforced hems, to be closed with buttons, or they were, as the name suggests, bands that ran in a tunnel to be tied in the front or the back.
Today, underwear waistbands are practically always elastic. They are typically different for different types of underwear, and also different for male and female underwear: waistbands for boxer shorts are thicker than those for men's briefs, and they are even thinner (and sometimes lacy) for women's panties. Since the 1990s, waistbands for name brand briefs often have the name of the manufacturing company on them.
Waistbands, sexuality and spanking
The waistband is symbolic for the access to the genital and buttocks region. Trousers are taken down by opening the button(s) and/or zipper and then pushing/pulling the garment towards the knees. Underpants and some shorts are taken down by inserting both hands into the waistband and then pushing/pulling the garment down. It is also possible to fondle one's partners (or one's own) buttocks or genitals by slipping a hand into the waistband without taking the underwear down.
The waistband is thus also the gateway to a bare bottom spanking. Even when the trousers are not taken down for a spanking, their waistband makes a good 'handle', e.g. to keep the spankee's rear in place or to pull the trousers taut over the buttocks.