Rigging is the arrangement of sails on a ship. Basically there are two kinds, "square-rigged", where the sails and spars (on a mast you have the wooden beams to which the sails are fastened. These are called spars), go in the direction left - right, and "fore-and-aft", where they go front-back (like most sailing yachts today).
The SPD Brigantine is actually more similar to a brig than a brigantine in it's rigging. Of course there were a lot of variantions, more or less from ship to ship, but in general, a brig has two masts, both square rigged, whereas a brigantine also has two masts, but a square rigged foremast and a fore-and-aft rigged mainmast.
For comparison, you can compare the brigantine Irving Johnson to the brig Lady Washington. The rigging on the SPD Brig is nearly identical with the latter.
Here are the names of the individual sails on this magnificient ship:
Above the main topgallant sail was occasionally a very small sail, called the royal.
The Boommainsail could sometimes be called the gaff, but in my opinion, the former is more correct. Especially since that is the most important sail on the SPD Brig (basically the one doing most of the job).
The "wings" rigged fore-and-aft (sails E & F) are called studding sails, and named after the sails to which they are fastened.