Chemist's is the shop where medicines are sold. It's only used in British English. Pharmacy is the place where a pharmacist works. In American English, this may be a shop or a part of a hospital. In British English it's used only to refer to a part of a hospital.
chemist's We need aspirins. Tomorrow I will go to the chemist's. She went to the chemist's but forgot to take the needed prescription.
pharmacy We need aspirins. Tomorrow I will go to the pharmacy. She's been working for the hospital pharmacy for five years.