Diamonds are used for premium items/actions; Such as constructing exclusive Buildings, Buy Collectibles, Skip task in a Quest, etc.
How to get Diamonds
Level Up: You'll get 1 Diamond for each level-up.
Buy with real money.
Diamond vault: Use a Small, Large, Elite, or Epic Diamond Vault (became available on the 26th of October 2012). The average amount of Diamonds per day for the Small Diamond Vault lays around 2/day. Do not expect to get 10/pay, or 40/day for the Epic Diamond Vault.
Burt Jackson: You might have seen a big truck on your city's roads driving around. You can buy a key for it to open the truck, which gives you a chance to get Diamonds from it.
Diamond Club: The more Diamonds you spend, the more you get back through the Diamond Club. This is not a real way of receiving Diamonds, as you'll only recuperate +-10% of your Diamonds.
Quest Completion: Some quests have diamonds as rewards.
Spin City: You get one daily spin for free from Spin City, and it can award diamonds.
Collect from the Mayan Temple. Up to 3 diamonds every two days.
The best way yet is to buy a Diamond Vault. Diamond Vaults are, apart from leveling up, the best certainty you get Diamonds every day, without interruption. However, they are extremely expensive, as you'll have a virtual unlimited supply of Diamonds, and if you put them in your inventory per accident, the timer starts to count from zero again. So you'll have to wait another 24 hours for the Vault to become ready for harvest.