When and for what health reasons should we break our fasts? (assume the person is not elderly or pregnant.)
Mishnah Berurah 550:4 says that one who is ill (choleh she'ein bo sakana, meaning that the illness is severe enough to make one bedridden, or it affects one's entire body) is exempt from fasting, and is actually forbidden to fast.
As the other answer mentions, a sick person is not obligated to fast. A regular fast day is a observed as a minhag nowadays, but even on tisha b'av (an actual chiyuv), the basic exemption for sick people still applies. On yom kippur, one may only break the fast if one will be in (life-threatening) danger otherwise. Specific cases should obviously be referred to a rabbi and/or doctor.