What could cause extreme pain at the base of the skull on the right side?

Muscle or headache. Neck strains - from poor posture, whiplash or vigorous exercise can cause muscle strains that you feel at the base of the skull (that's where some neck muscles attach). Certain types of headaches can do this as well. I'd try some Ibuprofen or alleve and see if it improves. If you start having blurry or double vision, nausea, or fevers, i'd see your doctor right away! cheers!
Occipital neuralgia. The greater occipital nerve runs inbetween the neck muscle tendon insertions at the posterior (occipital) skull base. It can get compressed and cause severe pains. First line treatment - motrin/advil around the clock for a week, muscle relaxation and gentle neck massages. If not better, try occipital nerve block. If not better soon, see your md for an examination to pinpoint the diagnosis.