In American English?? =)
A market is a store that may be a small grocery store or a collection of booths where specialty items are sold.
A supermarket is a very large grocery store that also sells other items, such as pet supplies, cleaning items, and sometimes has a pharmacy and/or a bank in it. There are many around here that also include a small medical clinic.
A chemist is a person who works in chemical research. (In Britain, this is also a term referring to a pharmacist, the doctor in a pharmacy or drug store.)
A pharmacy is a store in which you can have medical prescriptions filled, as well as where you can buy over-the-counter medicines and other health and medical supplies to be used at home.
A shopping center is usually a collection of stores in one relatively large building.
A department store is one huge store in which clothing and shoes are in different departments for women, men, boys, girls, small children, and babies -- as well as items for the kitchen, bathroom, etc. The department stores around here usually are at main ends of shopping malls, such as in this map:
Sears, JCPenney, Dillards, and Macy's are the department stores that "anchor" this shopping mall.
A shopping mall is similar to a shopping center, but usually is MUCH larger, and all the stores and the walkways are covered by one huge roof.