A drawer is a sliding compartment that fits into a dresser and is used for storage. You might keep socks in one drawer and t-shirts in another.
Dressers have drawers, and so do other pieces of furniture, including desks, bureaus, and kitchen cabinets. Your kitchen drawers might be full of silverware, while the drawers in your office are a jumble of pens, paper clips, and Post-It notes. Drawer comes from the verb draw, from its "pull" meaning — the original idea of the word was that it's a container that's "drawn" out of a cabinet.