To become "awakened", characters must meet certain prerequisites such as dipping under a certain health level. Simply hit the L1 and R1, or LB and RB buttons on the gamepad to perform an awakening action. Keep in mind that you'll need chakra to perform the action, which lasts only temporarily. Check them out below:
Note that some characters will simply emit a cool glow while others will have a full fledged animation to depict their transition to an awakened state.
In addition to becoming awakened, each of your characters can perform an Ultimate Jutsu that unleashes their full potential and allows them to devastate their enemies completely. Check those out here.