Slide out the drawer, and remove its contents.
Extract the screw or screws on the inside of the drawer faceplate, using a screwdriver, in order to release the remains of the broken handle. Save the screws and any washers around them.
Take the broken handle with you to a hardware store to match it with a replacement. Depending on the age of the furniture and the style of the handle, you might have to decide whether to replace all the handles if you cannot buy an exact match or settle for a handle that approximates the look of the broken part.
Line up the hole or holes on the back of the new handle with the holes in the frontplate of the drawer.
Insert the screws through the holes in the back of the drawer's frontplate, and twist them into the new handle with the screwdriver.