Gomoku is in basically a simple game. There is an empty board with 15 rows and 15 columns. You and your opponent (computer in this case) take turns. The player whose turn it is, places his/her marker (either an X or 0, chosen at the start of game; first player is X) to an empty cell on the board. The objective is to get 5 markers in a row either horizontally, vertically or diagonally.
Obviously it is possible (an necessary) to block other player's attempts to win the game by placing your markers at locations that prevent the opponent from gaining that position. The trick is finding the right balance between blocking opponent's attempts at victory and trying to win yourself.
The computer opponent is not exactly difficult, but it can beat beginners. Don't expect it to have great plans as to what it tries to do, but it is not exactly stupid either.
In this implementation of gomoku, if you make a mistake, simply press "Back" button of your browser and play your turn again. If, however, the computer does actually win due to the errouneus move, the win counter will include the win even if you continue the game.