The Forge allows you to enhance your cards using special Crystals, which you can earn in the Season's Reset.
This feature is a part of the Season's Rese event, and it only becomes available when you reach 3800 trophies. Once you get there, the Deck button will become "Deck & Forge" button. 

Tap on the "Deck & Forge" button and then, on the lower left corner tap on Forge.

There you will find your Forge Crystals, which can be used to level up cards or even unlock a new max level to characters that are currently maxed out!
In order to improve your cards using Crystals, you must first match the Crystal's rarity with the card rarity. This means you can only use Legendary Crystals to upgrade Legendary cards, Rare Crystals to upgrade Rare cards, and so on. To make it easier, we have made the Crystals in the same colors as the card's edges, which show their rarity level.

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 select a card by tapping on it, then check the requirements to improve that card on the right side of the screen.

So here's how it works:

You can temporarily increase 1 level of any of your cards!
The improved card will show a "clock icon" to represent that it has been temporarily upgraded. After the time is up, the card will get back to the original level.

You can also permanently increase the maximum level of a card!
Keep in mind that your card will not be automatically upgraded after that. You will only unlock a new maximum level, so it will be necessary to acquire more cards of the same character in order to upgrade it, ok?
Also, it is necessary to upgrade a card to its maximum level before being able to unlock a new maximum level.