An ache is discomfort that continues for some time. It is usually associated with a specific part of the body, such as a headache, a stomachache, a toothache, and an earache. After you exercise, the next day your muscles will probably ache. An ache is usually not extremely strong, so you can try to ignore it.
Pain is usually stronger, more sudden, and more difficult to ignore. You would feel pain when you cut yourself or hit your head on something. If you exercise and you injure yourself – break a bone or tear a muscle – you would feel a sudden pain.
We also have the expression “aches and pains,” which describes general and various physical discomforts. Your 90-year-old grandfather might complain about all the “aches and pains” he has at his age!
Hurt is a little different because it is usually used as an adjective or verb, not a noun. To describe an ache or a pain, you could say:
Hurt is also used to mean “injure”:
Finally, all three of these words can be used to refer to emotional pain as well as physical pain:
(heartache = emotional anguish)
(pain = emotional injury)
(hurt = upset, sad)