Incompatibilism is the position that determinism is incompatible with human freedom. Hard Incompatibilism is the position that determinism is incompatible with both human freedom and moral responsibility.
There are two kinds of incompatibilists, those who deny human freedom (a view that William James called "hard determinism,"), and those who assert it (often called voluntarists, free willists, or metaphysical libertarians - to distinguish them from political libertarians). As a result, incompatibilism is a very confusing term in the free will debates.There is a large group of thinkers who are hard determinists including Richard Double, Ted Honderich, Derk Pereboom, Saul Smilansky, Galen Strawson, and the psychologist Daniel Wegner.
Accordingly, there should be two positions for hard incompatibilists, but the current advocates typically say they are agnostic about the truth of determinism or free will. Some of them say that free will is merely an illusion.
Adding to the terminological confusion, indeterminism is also said to be incompatible with human freedom and moral responsibility, or at best provides an incoherent and unintelligible account of freedom.
If the proximate cause of our actions is undetermined - for example, the result of an uncaused quantum mechanical event in the mind - it would not be freedom of a kind worth having and we should disavow responsibility.
Besides hard incompatibilism, incompatibilists have staked out nuanced versions of the familiar positions with new jargon like semicompatibilism, and illusionism.
To see which philosophers hold which positions, take a look at our history of the free will problem.
Let's look at the taxonomy of deterministic positions and see where hard incompatibilism fits.
Semicompatibilists are narrow incompatibilists who are agnostic about free will and determinism.
Hard incompatibilists think both free will and moral responsibility are not compatible with determinism. Illusionists are incompatibilists who say free will is an illusion.
The old incompatibilism explains freedom. It cannot explain the will. Hard incompatibilism denies both freedom and responsibility. "Soft" incompatibilism gets us both free (random) and will (adequately determined).
The Cogito is genuine free will.