Kant discussed intuition (Anschauung) mainly in the Critique of Pure Reason. Its meaning has been extensively debated in the secondary literature. Part of the difficulty is that Kant does not directly discuss “psychological” matters in any detail, except for the very informal treatment in the Anthropology.
I had been used to considering intuition and thought in Kant as an inseparable hylomorphic unity, like matter and form in Aristotle. This would mean that when we speak of Kantian intuition, it is always as an abstracted partial aspect of a larger whole of experience. (See also Figurative Synthesis.) Recent reading has raised a question about this, though.
Most famously, Kant speaks of the intuition of a sensible manifold. This resembles Aristotle’s account of sensation as mainly passive, but complemented by and interwoven with more active processes (see Passive Synthesis, Active Sense). Kant developed this quite a bit more extensively than Aristotle did. Aristotle hinted at something like passive synthesis, but mainly used its tentative results (common-sense objects) as a provisional starting point. Kant tried to reach back further into the preconscious generative process. My favorite discussion of this is Beatrice Longuenesse, Kant and the Capacity to Judge. (See also Kantian Synthesis.)
According to Kant, mathematical construction, which produces an object and not just a theorem, operates on the basis of a kind of pure intuition not tied to sensory perception. This was the original inspiration for Brouwer’s mathematical “intuitionism”.
More broadly, I think Kantian intuition corresponds to the element of immediacy in experience, including what I have called feeling, as well as a kind of holistic summation of previous experience preconsciously associated with patterns preconsciously discerned in the current manifold. There seems to be a complex reverberation and mutual determination between immediate and mediate elements in experience. This appears both in the Kantian transcendental deduction (see Longuenesse, cited above) and in the Hegelian idea that immediacy is always “mediated immediacy” and thus never purely immediate. It also again reflects the fundamental hylomorphism of intuition and thought.
Something like Hegelian ethical Spirit or the Kantian transcendental is all mediation, in contrast to traditional views of spiritual or mystical experience as something immediate and unanalyzable. I take Kantian intuition, Brandomian sentience, and the main import of Aristotelian soul to be on the immediate side, but subject to the reverberation and mutual determination mentioned above. (See also What is “I”?; Psyche, Subjectivity.)
In contrast to Descartes and Locke, Kant famously rejected the idea of intellectual intuition, or passive reception of thought contents, just as he rejected the medieval notion of the intellectual soul. Anything “intellectual” would be on the side of thought rather than intuition for Kant, and thought for Kant always involved explicit, active development rather than passive reception. Hegel, Sellars, and Brandom take this as a starting point, and I think Aristotle would concur. (See also Subject.)