Eventually there will be an array of cards on the table which follow some pattern and the goal is to figure out the pattern. There is a dealer who makes up the pattern and does not tell anyone else. The pattern must be chosen so that each successive card depends only on the immediately previous card. For example, the cards could alternate between multiples of 3 and non-multiples of 3.
The other players are dealt 14 cards each and they take turns putting a card out on the table. The dealer says if the card is correct or incorrect. If it is correct, it goes in the "main line" of cards which obey the pattern. If it is incorrect, it goes on the side of the previous card in a "side line" so the players can study wrong plays as well as right plays. In addition, the player gets some penalty cards. So the practical goal is to get rid of your cards. When you know the rule, you do not say it out loud; you just play correctly to get rid of cards.
There are many other rules to make the game more interesting and to model other aspects of scientific research and there is a complicated scoring method. A complete set or rules are given here. However you should also read Gardner's article to get a full discussion of the analogy to scientific research.