I have a hard lump under a decayed tooth, this tooth has caused me severe pain in the past but has since rotted and decayed, what could this lump be?

Lump. Infected damaged bone. If you had an infection anywhere else that was spreading would you ignore it. This can actually be quite dangerous. Run, do not walk, to your Dentist MMedSciiately.
Chronic abscess. may be painless, but is sometimes the most destructive. If treatment of the abscess is not commenced immediately, the infection could spread. The result could be fever, swelling and severe pain.
Lesion. It is a lesion of some sort. Exactly what is hard to say without visualization and a radiograph. Appears you are long overdue for a dental visit. Please call a local dentist for an appointment.