What is the best way to remove and reuse wooden fence boards?
Flat prybar:  



(Almost self explanatory, but...)  

Just reach over the top, get a flat prybar between the top rail and the board, pry the top off, then pull the top down - the board may pull the nails out of the rail, or you may have to use the prybar again on the bottom.  This will leave the nails with the boards, drive them back out by hitting the points from the back, and pull them out with pliers or a nail puller.  

After the first one, you'll be able to easily get between the rail and the board top and bottom and it should go pretty fast.  

If the wood's rotten or very soft, the heads may tear through.  The first one is probably the one most likely to get mangled so pick an ugly one.  This isn't exactly restoration architect technique, but it should work.  

Another gentler way is to get a small hole saw just bigger than the nail head, and cut around each nail.  With a good hole saw and cordless drill this will actually go pretty fast.  This will leave holes in the boards but they will be perfectly round and easy to plug if need be.