The problem of akrasia is this: the incontinent person seems to know that what they are about to do is bad, and yet does it anyway. This looks contradictory. If they really know it is bad, how can they choose it? Socrates concluded they must not really know — wrongdoing is always ignorance in disguise. Aristotle thinks this saves the theory at the cost of the phenomenon.
Aristotle's resolution is subtle. He distinguishes between having knowledge and actively using it. The incontinent person possesses their moral knowledge in the way a person asleep or drunk possesses knowledge — it is there, but not operative at the moment of action. Passion overwhelms the capacity to bring general principles to bear on the particular case. The knowledge is not absent; it is temporarily overpowered.
The distinction between the incontinent person and the vicious person is crucial. The vicious person has bad values — they genuinely think that indulging their appetites is good and do so without conflict or remorse. The incontinent person has good values but fails to act on them, and afterwards feels regret. This is morally significant: the incontinent person can be reformed, because their practical judgment is sound; only their self-governance is deficient.
Aristotle's treatment of akrasia matters because it takes seriously the complexity of human motivation. We are not simply rational agents whose actions follow transparently from our beliefs. We are creatures with passions, habits, and competing desires, and these can overwhelm our better judgment. A moral philosophy that ignores this fact — that assumes we will do what we judge best — is a philosophy for beings more rational and less embodied than we actually are.
Akrasia is the central topic of Book VII, Chapters 1-10 of the Nicomachean Ethics, where Aristotle engages directly with Socrates' denial that weakness of will is possible.