Hume analyses what he takes to be the ordinary concept of causation - the idea of causation that people actually use - and breaks it down into three elements : (1) temporal priority of cause to effect, (2) contiguity (closeness in space and time) of cause and effect, and (3) necessary connexion. This last means that given the cause the effect cannot but occur; the cause necessitates the effect.
Hume re-analyses the concept of causation. In this sense : he retains the temporal priority of cause to effect and the contiguity of cause and effect. But his empiricism prevents him from keeping the idea of necessary connexion. This is because he bases all knowledge ultimately on impressions - sensory perceptions. And his key contention is that not only don't we but we cannot perceive necessary connexions between events or states of affairs, or whatever we take to be the terms of the causal relation. What would it be like to perceive a necessary connexion ?
Take an example. You pour water into a kettle. You put the kettle on a lighted gas ring. The water goes from cold to lukewarm to hot to boiling. You regard this as a causal sequence of events. Hume does, too : but he asks, where in that causal sequence did you perceive necessity ? Nowhere and at no time. All you perceived was a succession of changes in the heat of the water.
So necessity has to go. Hume replaces it with 'constant conjunction' : when A-type events precede B-type events, when A- and B-type events are contiguous, and when (crucially) A-type events are regularly followed by B-type events then we say that A caused B. All that we are actually 'given' in perception are priority, contiguity, and constant conjunction (regularity of joint occurrence). We may think that A necessitates B but all that we are empirically justified in saying us that A-type events and B-type events are 'constantly conjoined', they regularly co-occur.
This 'deconstruction' of the concept of causation is quite strong enough to support claims of moral responsibility. There are two sides to this. On the one hand, if the world (or our experience of it) did not exhibit constant conjunctions then no-one would know what the consequences of their actions would be. In sceptical mood Hume doesn't believe we do know. But for the purposes of ordinary life, we 'know' that a glass will (probably) break if we drop it onto a stone floor. There is enough de facto predictability to sustain moral responsibility. This is Hume outside his study.
On the other hand, unless certain constant conjunctions held in a person's character - a key term for Hume - we could not hold that person morally responsible for anything. If given a certain belief and a certain desire, a person might form just any motive, and if with a given motive a person might form just any intention, and if with a given intention a person might do just any action, then how could they be held morally accountable for what they do ? There is no connectedness between desire, belief, motive, intention and action sufficient to sustain moral responsibility. It's because people are not like this but have a Humeanly causal mental life (packed with constant conjunctions) and act in a Humeanly causal world (equally packed with constant conjunctions), that moral responsibility is possible and real.
Hume is a determinist in the sense that he thinks that causation is pervasive but then, he doesn't think causation is anything like what we take it to be. Humean causes don't make anything happen; they simply register constant conjunctions under conditions of priority and contiguity as spelt out above.
Hume, Treatise of Human Nature (1739-40), I.iii, 1-14 (esp. 14); I.iv.7.
Hume, Enquiry concerning Human Understanding (1748), IV-VII.