Virality game mechanic is where the game offers rewards to people that play it for spreading the word about the game and inviting other players to join. This mechanic is often used in Facebook games like Farmville, or online games like World of Warcraft where you can refer a friend or ‘Recruit a friend’. By doing this you get special rewards that are normally unobtainable.
Like in this image, it shows that with friends you invite on Facebook you can “Reach Super Rewards Early!”
This is the reward screen from World of Warcraft which lets the player that has managed to recruit a friend pick a rewards of their choice out of this list. Normally in order for the player to receive a reward a player has to help the recruited friend to reach a certain level milestone.
In order to implement this mechanic I would more than likely have to create an online game. For that reason I would include special rewards for people that play with their friends, for example an experience gain boost for both of the players.