What is an Awakening
Awakening allows your fighter to enter a heightened state where their power and the damage their abilities deal is substantially increased. Some characters such as Vegeta going Super Saiyan physically transform during an Awakening.
How to use an Awakening
An Awakening Gauge will build up around your fighter's portrait as they battle other characters and once it reaches 50% full you will have the option of holding down the R3 button (right analogue stick) to put your character into their awakened state. This will boost their damage as the gauge drains, until it's depleted and they return to normal.
What is an Ultimate Awakening
This is a second tier Awakening that powers up your character even more and can be achieved two different ways. You can either wait until your Awakening gauge is completely full which will be indicated by it glowing red and then holding down the R3 button to enter Ultimate or aternately, you can achieve Ultimate by using your first Awakening so its meter fills up to at least 50% again and then using it. Certain characters such as Goku who transforms into Super Saiyan Blue with his Ultimate Awakening will have a completely new look.