The goal of the game is to order unbroken row of five signs horizontally, vertically, or diagonally. You play by clicking with mouse on any empty field of the board. Then it is turn of Player 2 (in this case Computer). And then it is your turn again and so on.
Gomoku is an abstract strategy board game and is also called Five in a Row. It is traditionally played on a board with size 19x19. However, because once placed, pieces are not moved or removed from the board, Gomoku may also be played as a paper and pencil game. This game is known in several countries under different names.
The name Gomoku comes from the Japanese language, in which it is referred to as gomokunarabe. Go means five, moku is a counter word for pieces and narabe means line-up. You can read more on Wikipedia Gomoku page.