What does it mean if you have tooth pain that moves around?

Depends: Tooth pain that moves around can be from one tooth causing referred pain to other teeth. The pain can be from recession at the gum line exposing the roots of the tooth which would cause this situation, the remedy would be to see your dentist to diagnose the problem, .

Teeth or gums: You might have numerous areas of decay which will become sensitive if your mouth becomes acidic in areas of impacted food. You may have generalized periodontal disease (gum disease) where symptoms come and go depending on the progression of the condition. Anytime you have tooth pain, it is important to see your dentist. By the time you have pain, the problem may be fairly big.

More than that: You have more than one problem. Have your dentist evaluate what the real problems are.

Hard to say: Hard to say without more information. Could be problems with multiple teeth or gum areas. Could be bruxism causing pain. Sometimes there are non-dental causes. Best bet is to see dentist for evaluation.