What is this small bump under the bottom lid of my eye; is it a stye?

Yes: Yes, this most likely is what is commonly called a stye. See your doctor to know for sure. Don't put your eyes at risk.

Stye: Warm compresses (wash cloth with hot water and hold it on the stye for 5-10 minutes, reheating as needed), or surgical incision and drainage, or steroid have been the only effective treatments demonstrated get rid of a stye. A stye is due to the orifice of an oil gland (mebomian gland) in the eyelid margin becoming inflamed, and not an infection, so antibiotics are not indicated.