A supermarket is a large self-service shop selling foods and household goods.
Department store is a large store often on several floors and there many departments where you can buy different goods.
Hypermarket is a huge supermarket often out of town and there you can buy everything from food to electrical household goods.
Shopping mall is a building or several buildings with many different shops. Usually there are food court or restaurant, some entertainment and so on.
Corner shop is a small shop which sells everyday items, usually serving a small area and open long hours.