Reciprocity means that your "permit" will be recognized in the state of interest. The laws of that particular state apply while you are there.
It's done on a state-by-state basis. Right now, there are about a dozen or so states that will recognize a Washington CCW permit (packing.org is a great website that has all the info for each state).
Washington recently passed a bill allowing for reciprocrity with other states with "similar" CCW issue laws to those in WA. Unfortunately, the bureacrats are being strict with their interpretation, so WA-FL reciprocity hasn't happened yet.
If you travel a lot out-of-state, I would recommend getting a non-resident Florida permit. They are easy to get, and with one, you'll be "legal" in almost 30 states. With a Florida non-resident permit, you'll be able to drive all the way from WA to FL and remain armed, so long as you pick the right route.