Diamonds are the IAP currency of the game. They can be randomly found during a Fever Mode (1 in 3 base chance of finding one), obtained as a reward for doing Isabelle's quests (varies) or bought with real money.
Diamonds can be used to speed up tasks (amount varies depending on time left), buy rare Potions from the Item Shop (the price will slowly increase and cap to 91 Diamonds), buy more Coins (the mall's level determines the diamond-to-coin ratio and is capped around level 50, so it is best to reach at least level 50 before buying) and buy more Workers (250 for the first one, then 500, 750, and so on).
Diamonds are not required for players to enjoy all of the game's features, meaning it is entirely possible for a patient enough player to get through the whole game without any purchase. Bear in mind this is a game that requires a lot of patience.