A dresser is a piece of furniture with several drawers that's used for storing clothes or other things. You might have a dresser in your bedroom that's full of sweaters.
In the US, the word dresser almost always refers to a tall, upright type of furniture that's fitted with sliding drawers. You can keep your socks in a dresser, or put a dresser in your kitchen and keep tablecloths and silverware in it. If a person is called a dresser, she either works with theater actors, helping them put their costumes on, or dresses in an unusual or distinctive way: "She's such a colorful dresser!"