Some awesomes are not born with their powers or train hard to gain abilities, some people receive them for reasons or no reason at all. Power Branding is the physical sub-atomic reaction in which someone gains powers.
Not to be confused with a quickening, a branding gives the awesome their power. A branding can be anything from a lightning bolt from the sky to a terrible accident in the lab, but when it happens, the person is changed for good.
There aren't that many awesomes in The Revengerists, but characters like Righteous Oxide, Harbjar, and A Vampire went through a sort of branding and I guess Hotknife. It is important to stress that self-made awesomes were never branded, guys like Harbjar, Oddjog, and Dr Evilus are not Branded Awesomes. Certainly not Adobodo or Dr Tasty.
A power branding is not an infallible event, they can be manipulated. Sometimes an awesome receives different powers or none at all. Branded individuals can be possessed like Hacksploiter was by Dr Tasty. Dar'kal Bloodrazer once stopped a branding of a potentially extremely powerful awesome by cutting off the supply of power at the right moment, but this was retconned when it was revealed Han Dwaving was diverting the power stream the whole time.