To "level up" an emoji's special power, you need to get a duplicate of that emoji, usually by buying a silver or gold box, or otherwise earning the emoji. When you "level up" an emoji, its special power gets stronger.
To "power up" an emoji's special power means that while your hero emoji is in play, you clear instances of the hero emoji from the play board. As you clear instances of the hero emoji during play, you gain points, which translates into filling the power bar for the hero emoji. When the power bar is filled completely, the emoji is "powered up" and its special power is active. At that point, you'll see the hero emoji in the lower left area of the screen bounce lightly up and down (it is easy to miss during play.) When you see the emoji bouncing, tap on it to invoke its special power and use it.