Diamonds are the premium currency of Forge of Empires. You do not need to purchase Diamonds to play the game. However, with Diamonds you can get useful benefits that can for example speed up your progression in the game. Diamonds can be bought with real money, but can also be acquired through some quests and community contests.
To purchase Diamonds, simply tap/click the + button next to the place where your available Diamonds are shown in the top right corner of the screen.