Brunette Girl Wearing Black Stockings On The Couch

A garter belt (AmE), or suspender belt (BrE), is the most common way of holding up stockings. It is a piece of lingerie worn around the waist like a belt which has "suspenders" or "stays" that clip to the tops of the stockings to hold them in place.
"Stay-up" technology is the second most common means of support. The inside of the top of the stockings has a band (typically silicone) of elastic of highly tractive material that resists slipping down the thigh. This is considerably less reliable in the sense of assuring that the stockings do not fall down, and depends a great deal on the thickness of the thigh. Further, the elastic can be uncomfortably tight, leaving red marks and possibly aggravating varicose veins.
A garter is the least common means of support. It is slipped over the top of the stocking to hold the stocking by essentially clamping it to the leg. These are the garters typically seen at weddings. They have much the same disadvantages as "stay-ups".
Traditionally, panties were worn underneath garters, and lingerie is still advertised and photographed in this way in catalogues and on websites. However, some people wear panties over the garters. If the panties are reasonably tight, this pins the garters to the body, making going to the bathroom or erotic play easier.