Can someone tell me exactly what "double clutching" means? I've asked a dozen people and each one has a different answer.

It refers to declutching as you go through neutral during a shift. Its a carryover from the old non-synchro days when you had to match revs with the trannie to get it into the next (usually lower) gear. Often combined with heel-and-toeing (q.v.) but not the same thing at all.