Ok, lets go back.
At the very beginning there was two brothers, a big brother and a little brother.
The big brother has the ability, as far as we know, to steal or grant quirks to others. This is know as All for One.
The little brother was quirkless. His big brother, feeling bad for him, gave him the ability to stockpile power.
But the little brother was actually not quirkless, he just had a quirk that had no physical manifestation. His original quirk was that he could pass down his power to someone else. This in itself was useless, but when he got the stockpiling power quirk and they both merged, forming One for All. This quirk allows the user to stockpile power and nurture it, and then pass it down to the next user.
The big brother was using his powers for evil, so the little power wanted to stop him, but as he didn't have enough power to do so in his lifetime, he passed on his ability to another person, in the hope that sometime in the future One for All could be strong enough to stop All for One.
All Might, who is aprox. the seven or eight user of One for All, believed to have enough power and did fought against All for One, and in the process got the injury that we see in the first chapter. But All for One escapes and survives, so All Might is not strong enough anymore to be the one to defeat him, so he looks for a successor that could nurture One for All´s power to its full potential and finally take down All for One.
For all we know, the big brother, who somehow survived to this day, is not aware that his little brothers power is living on in Deku nowadays.
