Crystals are a new currency similar to Diamonds. The difference between them is that Diamonds must be paid for in real-world money; Crystals are free. You can earn Crystals from daily login rewards or by participating in events.
Most features that previously could only be paid for in Diamonds, such as buying champions or reforging Gems, can now be paid for in Crystals. But you can still use your Diamonds for these features as well! Diamonds can be traded in for Crystals on a 1:1 basis. If you don't have enough Crystals to pay for something, you can make up the difference in Diamonds.
For example, let's say you want to buy the champion Adela. She costs 999 Crystals, but you only have 700 Crystals. You also have 400 Diamonds. If you tap the Purchase button on Adela's portrait in the Store, you will be asked if you want to pay 299 Diamonds in addition to your 700 Crystals. If you accept, you will be left with 0 Crystals and 101 Diamonds, and Adela is yours!
This also works if you don't have any Crystals at all, effectively allowing you to buy champions and other services for their original Diamond price. So don't worry about having to collect Crystals in order to buy something you'd been saving your Diamonds for; you can still pay for Diamonds and use those exclusively if you want.
Note that you can only trade Diamonds for Crystals; you cannot trade Crystals for Diamonds. Also, you cannot choose not to use Crystals to pay for something. The game will always prioritize Crystals over Diamonds when deducting your currency.