I broke off a piece of molar but it didn't start hurting until 6 months later. now it's sensitive and throbbing. also pain in my jaw what could it be?

Infected tooth: You have let the tooth remain untreated and the bacteria have likely reached the vital tissue. This means that the pulpal tissue which contains the nerve and blood supply is either dying or dead. The breakdown products of this have to go somewhere and this is usually the bone and gum tissue. You may be able to save the tooth, or have it extracted and get an implant and crown. See dentist ASAP.

See any dentist ASAP: A tooth abscess won't go away without treatment and the infection may spread to your jaw, cheeks, sinus and to other areas of your head and neck. Dentist will treat a tooth abscess by treating the cause and getting rid of the infection. See your dentist ASAP.