What is this tremendous pain in the right side of my neck, just below the base of my skull?

See your doctor. If you have severe neck pain, you should see your doctor right away.
Trapezius strain. The cause depends on the presence of other symptoms that go with your pain - presence of headache, radiation of pain, its character, presence of fever, visual symptoms, other body symptoms. If the pain is localized and is associated with head turns and no other symptoms present, it may be a benign muscle spasm, like trapezius strain. If in doubt, have a doc look at you.