Certain characters, such as Goku, have special power-ups. You can try this out in training mode. Your Spirit gauge must be at 100% to use Goku's or another character's power-up. Then press L + R. Goku will be blinking red. He is using his Kaio Ken (which is his power-up). There are other characters that have power-ups. Also, different levels make a difference. If you are testing out Goku's Kaio Ken, then first try his level 1 form.
Note: Be careful when using these power-ups. As you use it, your life slowly drains away, giving you a very strong disadvantage. You will not die when you use the power-ups; it will leave you some life, but not much. If you get hit, you will die.