The area under my left eye is red, swollen and tender to touch. i do not believe it is a sty or pink eye - what could this be?

Inflamed skin: If skin is red, swollen and tender to touch it is inflamed. The most concerning cause of inflammation is infection, so keep an eye out. Chronic irritation from rubbing or watering eyes can do it. Bug bites can do it. Sometimes allergies can do it, although these usually itch more than hurt until they get infected. You might also be in the early stage of a stye. Keep an eye out for a bump on eyelid.

Depends on where: As you suggested, chalazions or styes, are a common cause of inflammation of the eyelid. Other less common causes include a tear duct infection (dacryocystitis-usually below inner eye lid) or a cellulitis (infection of the skin). Both of these issues require topical and/or oral antibiotics and should be evaluated by an ophthalmologist. Of course there are other possible but more rare causes.