Shougi is Japanese chess. It's played similar to regular chess with 2 exceptions: you can 'promote' your pieces by flipping them over once they've crossed a certain point on the board, and if you capture an enemy piece, you can place it on the board as one of your own during your turn. Hence, there's quite a bit of strategy involved and the risk of sacrificing a piece to get a better position is higher.
Mahjong is akin to a card game played with tiles instead. You are trying to make either 3 consecutive numbers or 3 of the same tile of the same pattern set. The tiles are arranged so that you can only make 4 sets; the last one is 2 pieces only and you have to get 2 matching tiles in order to win.