Compatibilism holds that free will and determinism were never actually in conflict, and that the whole dispute rests on demanding something of freedom that nobody needs.
Freedom as absence of constraint
Hume put the classical version bluntly. To be free is to act according to the determinations of your own will: to do what you want, unhindered. The opposite of freedom is not causation but constraint.
There is an obvious and important difference between handing over your wallet because you decided to give to a collection, and handing it over at knifepoint. Both are caused. Only one is coerced. A prisoner is unfree, a person under hypnosis is unfree, a person in the grip of a compulsion is unfree, and none of that has anything to do with whether physics is deterministic. Compatibilists say this ordinary distinction is the real one, and it is exactly what our practices track.
Wanting what you want to want
Harry Frankfurt refined this with a point about the structure of desire. An unwilling addict wants the drug, and also wants not to want it. A willing addict wants the drug and endorses that desire.
Both are moved by the same craving, but only the first is alienated from what moves him. Frankfurt argued that freedom of the will consists in this identification: your will is free when the desires that move you are ones you reflectively endorse. Nothing here requires an undetermined universe. It requires a certain relationship between you and your own motivations.
The problem of alternative possibilities
Frankfurt also attacked the assumption that responsibility requires being able to do otherwise, with a thought experiment. Imagine a neuroscientist who wants Jones to vote a certain way and has implanted a device that will intervene if Jones shows any sign of deciding differently. As it happens, Jones decides that way on his own, and the device never fires.
Jones could not have done otherwise, since the device guaranteed the outcome. Yet he seems fully responsible, because the actual chain of events ran entirely through his own deliberation. If that verdict is right, then what matters is the source of the action, not the availability of alternatives.
The standard objection
Critics call this a redefinition rather than a solution. Kant dismissed compatibilist freedom as a wretched subterfuge: a clockwork that turns itself is still a clockwork. If the desires you endorse were themselves installed by genes and upbringing you did not choose, then endorsing them is one more determined event, and the deeper worry about origination is untouched.
Compatibilists answer that the deeper demand is incoherent, and that a freedom nothing could satisfy is not worth mourning. Either way, the argument has consequences for how we treat each other.