Hume defines belief as a lively idea related to or associated with a present impression. When you see smoke and immediately think "fire," the idea of fire is enlivened by its connection to the vivid impression of smoke. That enlivening is belief. It is not a separate act of judgement; it is the transfer of vivacity from impression to associated idea, caused by the habitual connexion between them.
Belief comes in degrees, corresponding to the strength of the associative habit. If fire has always been followed by smoke in our experience, the belief is near-total. If the conjunction has been intermittent, the belief is weaker — what we call probability. Probability is not a feature of the world; it is a feature of the mind's habit-strength, measured by how often the association has held in past experience.
This account is deliberately anti-rationalist. We do not form beliefs by consciously reasoning from premises to conclusions. We form them by the automatic transfer of vivacity through habitual associations. Reason is a post-hoc commentator on a process that has already happened. The implication — which Hume draws without flinching — is that our most confident beliefs about the world are, at bottom, sophisticated habits rather than rational achievements.
The theory of belief as vivacity and the probabilistic treatment of induction appear in Sections V–VI of the Enquiry. Hume's account anticipates later Bayesian theories of credence, though his "vivacity" language is harder to formalise.