In Hume, Kant, and Husserl, “intuition” refers to the immediate sense or understanding apprehended in apprehending something – the original acquisition of data – as distinguished from those cases where we later apply reflective thinking and reasoning as a method to analyze and draw further implications from what we understand.
Empirical intuition is sensual or sensory. In empirical intuition we apprehend sensible qualities of things simply by feeling them. In Hume’s account this is true of sensory impressions of external objects impinging on us and the feeling of internal affections in ourselves.
Intellectual intuition, in contrast, would then refer to a non-sensory, intelligible, or categorial apprehension of the logical, intellectual, or intelligible meaning of something, immediately, without requiring an application of reasoning from arbitrary definitions to arrive at that meaning. But isn’t this merely a reversion to pre-modern, pre-critical dogmatic philosophy, simply asserting the meaning of things without reasoned justification? Husserl argues that phenomenology avoids dogmatism by providing a method of descriptive justification: phenomenology provides methodical descriptions of intuited, essential intellectual structures in experiences, which yield the meaning of experiences; it does this in extraordinary detail, such that others can follow the same method and discover the same phenomenological evidence (Evidenz) of intellectual intuition, in their own experience.
Kant’s argument for transcendental categories of understanding (space, time, substantial entity, causality) is that they are forms of intellectual intuition rather than empirical intuition. That is, like empirical intuition of sensory qualities, the transcendental categories of understanding do not depend on reasoning in order to establish their legitimacy. We simply find them ‘there’ in any objective apprehension, even though they appear to be supplied by the human condition, rather than supplied by the nature of the objects out there.
Intuition for Husserl is similar, except he does not seem to give any doctrine of categories a priori. For Husserl, concepts are not merely definitional tautologies made up in human minds untethered from reality; rather, intellectual-conceptual (or categorial) and empirical-sensory intuition derive from the immanent intelligibility of experience of the world, and it is the job of phenomenological description to give radical accounts of experience in order to reveal this.