We have all had this experience: we hear of a complicated social or personal event, and we think inwardly, "that's wrong!" A co-worker tells us an embarrassing private story about another co-worker; we hear on the news that the number of children in poverty has increased; we read about a mining company that has dumped toxic chemicals into fishing rivers for years. And we have a moral intuition about what we hear -- not only a quick judgment about its goodness or badness, but also a sketch of reasons: "She shouldn't violate Frank's privacy that way"; "so much suffering of the innocent is awful"; "how can a company have such disregard for the people whose lives depend on those rivers." In other words, we have intuitions about complex situations that evidently rest upon some kind of reasoning -- but we haven't deliberated about the case.
What is the nature of such an intuition, and what kind of cognition does it represent?
As I've sketched it out, these "intuitions" are cognitively complex -- not just an inward "ugh!", but a sketchy representation of facts, assumptions, and relevant principles or rules. Our thought processes have somehow organized the description of the event into a story of sorts, along with some collateral judgments or principles about how people and institutions ought to behave. In the moment of intuition, we are also involved in judgment; and judgment involves something like reasoning and analysis. And yet the intuition itself is revealed in a moment -- not as a developing piece of analysis and deliberation, but an apparently instantaneous moment of moral perception.
If this description is phenomenologically correct, then moral intuitions are a feature of human judgment that involves complexity in something like the way that sizing up an auto accident occurs for the experienced investigator -- a quick cognition of the likely speeds and directions of the vehicles, the evasive action that appears to have occurred (skidmarks), etc. Following Kant, we might call this an act of "apperception", happening below the level of consciousness but bringing to bear quite a bit of analysis, knowledge, and principle in the construction of the resultant perception. And we might refer to an individual's overall set of interpretive frameworks as his or her "moral sensibility". Our moral sensibility provides us with a set of framing possibilities within the context of which we can begin to understand and represent the complicated human situations we encounter.
Now we can give a bit more content to the roles that moral analysis and principle play in this story. We are more or less forced to hypothesize that the person possesses a mental process of moral / social cognition that assembles a representation of a situation, including incorporation of a background set of principles or interpretive rules as well as a set of facts about the case, that eventuates in a morally tagged picture or narrative. The person is able to dig down into some of the underlying architecture of this picture if pressed; this accounts for the fact that we can give some analysis or explanation of our moral intuitions when asked, "why do you think this situation is wrong?". And the principles or rules that play an evaluative role are themselves learned through some concrete process of social development. (Though it is interesting to consider whether there might be a component of social cognition that is hard-wired through our evolutionary history; Alan Gibbard, Wise Choices, Apt Feelings: A Theory of Normative Judgment).
So one's moral intuitions are not grounded in something like "direct apprehension of moral facts"; rather, they are the result of a complex, conditioned, and fallible process of sub-conscious reconstruction of the circumstances of a case. And for this reason we might suspect that there will be significant differences across individuals and across cultures in the contents of people's moral intuitions about cases.
(It is worthwhile to contrast the idea of a moral intuition with John Rawls's idea of a "considered judgment". Rawls's idea captures the conception of a full, deliberative consideration of a case in detail, considering all the relevant facts and principles, the fit between a given judgment and many other judgments we make, and a host of other constraints of coherence. This picture is one of full, transparent moral reasoning and deliberation. The account of moral intuition just provided is non-deliberative but not for that reason non-rational.)
(This treatment of moral intuitions converges somewhat with Malcolm Gladwell's Blink: The Power of Thinking Without Thinking.)