Following my run last night, I later noticed that I have what feels like a muscle pull. Inner thigh, way high up. It mostly hurts when I walk and swing that leg forward. What is this muscle? Will it heal fast? Should I try to bicycle tonight or give it a rest?

Most likely your hamstrings... Take a rest until it heals.

Hamstring? I thought the hamstring was in the back of your leg. This is inner thigh, almost toward the front of my leg, and it feels like the pull goes partly up into the groin area.

Ok here are the muscles you could of "pulled" the gracilis muscle, adductor longus muscle, startorius muscle,pectineus muscle. Also the hamstring is composed of many smaller muscles stated above..