What would make your teeth hurt ?

Lots of things: Hot or cold sensitivity, dental decay, sinusitis, etc. See a physician or dentist if worried.

Oh so many things: Most commonly, infection of tooth and/or surrounding tissues. Can also be from fractures, parafunctional habits (clenching, grinding), gum recession w root exposure, response to medications, and many other reasons. Please see your Dentist ASAP to determine the source of your discomfort and the appropriate treatment.