The word intuition refers to two closely related but different things: a mental process and an example product of that mental process. It is easier to define the product, and then use that to define the process.
An intuition is an idea about something – a proposed fact – that pops into mind from no obvious source and for no obvious reason. It is not unlike a guess in that it is a proposition without sufficient grounds for concluding it is true. Another similar concept is a hunch: an idea about something that is not certain, but accompanied by a feeling of certainty.
Intuition, the mental process, is what generates intuitions as per this definition. Please note that the term intuition, in the colloquial sense[*], does not really have any status in the science of psychology, which tends to be much more specific in talking about unconscious processes that put thoughts into our conscious minds. Intuition is thus a phenomenological term: it describes a mental phenomenon from the vantage of experiencing it (as opposed to what it "really" is, whatever that might mean, or whether it is actually in some sense multiple other phenomena). But, ironically, since the experience of intuition is that of not knowing where an idea came from or the grounds one has for having it, intuition (the process) is something of a "black box".
When someone attributes an idea to intuition, they are effectively saying, "There is some unconscious part of my mind which has put together the information and evidence I (perhaps also unconsciously) have, to extrapolate or interpolate to come up with this conclusion it just dumped into my consciousness, unsourced, unjustified, and unexplained."
This should not be construed as saying that the grounds for an intuitive idea are unknowable, merely initially unknown; introspection often can dig up the details that lead a person to their intuitive conclusion.
Also, please be aware that intuitive, the adjectival form, is used in certain counter-cultural circles to describe someone who claims to possess ESP or other "psychic" abilities. When you see the phrase intuitive healer it probably does not mean a doctor who makes great diagnostic guesses, but rather someone who is offering supernatural approaches to remedying proposed supernatural ailments.
[* The term "intuition" is used in the Jungian (analytic psychology) tradition, but to mean something technical and subtly different than the colloquial sense of "intuition".]