A board press is a partial rep bench press, where someone places either a 2, 3, 4, or 5 board onto your chest, you lower the bar to the board, let it sink slightly, and press it back up. The boards themselves are made out of 2" x 6" 's, and are 24" long, with a 6" handle on one of the boards.
It's commonly done in PL training to train different portions of the bench.