The right headlight on our 2010 Toyota camry is dimmer than the one on the left and I think eventually goes out and the wiring harness connector is very hot. Any ideas what's wrong?

 

Check the connection for corrosion and other damage.

A bad electrical connection has more resistance to the electricity moving through it. As a result the bad connection will heat up much like a filament electric heater or electric stove burner although not this hot of course.