"Going Gin" means winning a hand with no cards left in your Deadwood. This means that you hand of ten cards must consist of no deadwood, only a combination of melds. As the tutorial says, "there are two types of melds - runs and sets." A set is when you have numerous cards of the same number (for example the 5♥, 5♠ and 5♣). A run is when you have consecutive numbers within the same suit (e.g. 6♥, 7♥ and 8♥). When you pick up the last card you need for going Gin, but before you discard your 11th card, remember to press to knock, and then to select the card you wish to discard.
For a full example, a hand that would win a round by going Gin might be:
2♠, 2♥,and 2♣; 5♥, 6♥, 7♥and8♥; and J♣, J♠, and J♦.
A gin hand cannot be beaten, and will earn you 25 bonus points, plus the entire deadwood in your opponent's hand.