I dropped a board on my big toe on the joint. it has been a week. the blood under the nail has gone from a real blue to purple black. infected?

Maybe...: It's possible, but may just need to be drained or have the nail removed completely to let a new nail grow back in 6-8 months. If it is still throbbing after 3 days then may be infected. See a doctor right away. Recommend an x-ray to make sure nothing broken.

No infection: It's called a subungual hematoma and will grow out with the nail in most cases.