The Forge Crystals are a type of currency within the game, and you can get them from your Season's Reset rewards. 
The Crystal types are:
Common Crystal - used to improve common cards.
Rare Crystal - used to improve rare cards.
Epic Crystal - used to improve epic cards.
Legendary Crystal - used to improve legendary cards.
Magic Crystal - used to unlock a card's new maximum level.

You can use them to improve your cards as following:
- Temporarily increase 1 level of any of your cards. Imagine you love to play with your level 7 Pirate, but you don't have enough cards to upgrade it to level 8. You can use crystals to upgrade the card to level 8 and play with it for 24 hours! After the time is up, the card will get back to level 7.
- Permanently increase the maximum level of a card. So imagine you have fully upgraded Dwarf card, and the maximum level is 13. With the Forge Crystals, you can unlock a new maximum level for your card! This means that the Dwarf's maximum level will be 14, and you will be able to acquire more cards and permanently upgrade it with gold!

Magical, isn't it? ;)