Immanuel Kant viewed the problem of free will and determinism as an antinomy (an apparent contradiction) that arises from looking at the problem dialectically and from two standpoints, the first theoretical reason, the second practical reason. From the standpoint of theoretical reason, human actions are phenomenal events occurring in the natural world and are therefore completely determined by natural physical laws. From the standpoint of practical reason, however, actions are noumenal events that result from a free will that deliberates between alternative possibilities, evaluates them, selects one, and thus acts freely by self-determination. Humans are determined when viewed (theoretically) from a third-person perspective as an object, but free when viewed (practically) by the "self" from a first-person perspective as a subject. These two standpoints correspond roughly to to David Hume's fact/value dichotomy, and more recently to Wilfrid Sellars Scientific Image and Manifest Image. It is sometimes argued that Kant was a dualist describing two "worlds" - the one phenomenal and the other noumenal. Kant's Idea of Freedom antinomy was in many respects a response to Hume, which Kant claimed could provide a defense of moral responsibility. But it has satisfied very few philosophers. Now determinists deny both freedom and moral responsibility, while compatibilists generally assert a special form of free will (a chain of causes in the mind) that is compatible with determinism and allows them to defend responsibility. But note the curious fact that all the participants in the free will debates are in basic agreement with Kant that there exists an Idea of Freedom, even as some of them deny that there is something phenomenally and physically real corresponding to the Idea and others redefine the meaning of the term "free will". Most compatibilists (and even some determinists) admit that on introspection they find that (despite their refined theoretical positions) as a practical matter they continue to deliberate over alternative possibilities, they continue to act on an Idea of Freedom. Compatibilists argue that determinism is compatible with human freedom, and that indeterminism is not compatible or at best incoherent. If our actions are indeterministic, they say, we cannot be responsible for them. (This is a critical part of the standard argument against free will). They feel (correctly) that there must be a deterministic or causal connection between our will and our actions. This, they say, allows us to take responsibility for our actions, including credit for the good and blame for the bad. In two-stage models of free will, the indeterminism is limited to the first stage, where alternative possibilities are generated, and the second stage is an act of self-determination. Two modern philosophers who have tried to make sense of the obviously contradictory (if not, in principle, irrational) standpoints in Kant's Idea of Freedom antinomy, are Christine Korsgaard and Dana Nelkin. Both these philosophers focus on the "deliberative aspects of practical reason, which require the existence of alternative possibilities to deliberate in an act of self-determination.
Note there is also incompatibilism. There are two kinds of incompatibilists, those who deny human freedom (usually called "hard" determinists) and those who assert it (often called voluntarists, free willists, or metaphysical libertarians - to distinguish them from political libertarians).
The first compatibilist was Carneades (214-129), the great Skeptic.