All celebrities are famous people, but not all famous people are celebrities. The word "celebrity" is derived from celebrate, meaning a person that is celebrated. They are followed by news reporters, you see them in television, they have fans. Someone is famous because they are well known within a particular community for some achievement. For example, within the medical community Dr. James (not a real person I am just making this up) is famous for inventing a medical device. He is not a celebrity since, other than a possible news article about the device, no one is regularly following him, he has no fans and is not continually in the news. We sometimes call friends and acquaintances celebrities as a term of affection or encouragement, even though everyone knows they are not celebrities.

famous : somebody who is known or maybe well known by many people; it could be present or past, known for good or for evil. celebrity: somebody who is 'celebrated', meaning that he did or does something so special, that many people will repute this person as very,very special and would try to get near him or her in many ways. Sometimes even evil people have got 'celebrated' during history. This term mostly stick to actors, singers, sometimes musicians; declaring someone as 'famous' is more or less under rational control, declaring somebody a 'celebrity' may well escape rationality, especially if you add in the equation all the wealth generated by the 'celebrity' shows. In other contexts /celebrity/ might mean: a recognized authority in a particular field of knowledge (or other subject).