A Referral Code is a feature in My Singing Monsters that gives 5 free Diamonds to both you and another player, allowed only once per account. It's intended to be a special thank-you for the person who introduces you to the game, although it can be used for someone else, by inserting their Friend Code in the Settings menu.
Since entering a Referral Code gives Diamonds, you might see messages in various places which promise a larger amount of Diamonds if you use a specific code, rather than if you use anyone else's code. These messages are lies, and are trying to trick you into using your only Referral for the benefit of whoever posted the message.