First of all, I'd like to apologize in advance that most of this will be Fi (and more tentatively, Ti) from an Ne perspective. While I feel that I can occasionally peer over the fence into Thinking territory, I remain pretty Sensing challenged. This also re-iterates a lot of what others have said, but I was trying to gather up various thoughts into a single post.
How Fi is like Ti
To paraphrase Jung "everything that is true of Ti is true of Fi, substituting Thinking for Feeling." It may be hard for Ti-users to fully appreciate, but I think there is a lot of truth there.
Both Ti and Fi tend to work from universal principles, tending to ignore surrounding prevailing standards. For Ti, those principles are founded in logical objectivity (although in terms of one's own theoretical system). For Fi, those principles are based on ethics, humanity and aesthetics (not that Ti lacks aesthetics). Neither Ti nor Fi place much emphasis on status or authority. In fact, both try to find the universal in the essence of the specific.
Both Ti and Fi value staying true to the essence of things. In Ti's case, it emphasizes staying true to things with logical consistency and precision. Fi demands staying true to things on the level of feelings, ethics and/or aesthetics.
Fi values staying true to one's self, one's nature, and the universal qualities of humanity. Ti values staying true to logical principles, precision and conceptual elegance.
Fi and Ti both tend to stand up for the importance of the specific reality. Fe and Te care more about general organizing principles applied consistently (and often externally), while Fi and Ti both are about adjusting understanding precisely in order to capture the underlying essence of the individual case.
Ti does have some advantages over Fi, since it is, in part, objective. Thinking, whether introverted or extraverted, more easily allows for external evaluation and validation. Logical theories and concepts can be detached from the people involved, and be both perceived and weighed for truth. Feeling doesn't have same objective measures.
The Nature of Fi
Jung states (Psychological Types, p387), "It is extremely difficult to give an intellectual account of the introverted feeling process, or even an approximate description of it, although the peculiar nature of this kind of feeling is very noticeable once one has become aware of it." Fi tends to elude verbal and logical descriptions, much to the frustration of both the Fi-user and his or her audience. This also hampers Fi-users in debate, since they are not always able to clearly explain and defend their deep understanding of the issue at hand.
Jung continues, "Conversely, with the introverted feeling type, feeling attains an abstract and universal character and can establish universal and permanent values." Introverted feeling forms principles based on those things that bind all of us together as human beings. It tends to respect the personhood, values and moral agency of the individual.
Introverted Feeling is also about harmonizing one thing to another. Actions should be harmonized with principles, deeper purposes and aesthetics. One's daily life should harmonize with one's underlying values. One person can harmonize with the feelings of another to better understand them. Fi can be rather exacting in this respect, leading to an kind of uncompromising and demanding perfectionism.
Fi is the judging function closest to the unconscious, so tends to communicate by means of feelings and primordial images.
Fi imparts value judgements before logical understanding. Therefore Fi-doms may understand something as essential, important and right long before they can defend why that is the case.
More Jung on Fi
Jung is refreshingly negative about aspects of Fi—none of the MBTI-style affirmation for him! He writes:
Jung p387 said:
[Introverted Feeling] is a feeling which seems to devalue the object and it therefore manifests itself for the most part negatively. The existence of positive feeling can be inferred only indirectly. Its aim is not to adjust itself to the object, but to subordinate it in an unconscious effort to realize the underlying images.
In a sense, introverted feeling wants to find the archetype that lies beneath, that captures the deeper essence. It seeks to relate the specific to the universal within.
Later on page 387 he says "[Fi] comes out with negative judgments or assumes an air of profound indifference as a means of defence." He continues:
Jung p387-388 said:
The primordial images are, of course, just as much ideas as feelings. Fundamental ideas, ideas like God, freedom, and immortality, are just as much feeling-values as they are significant ideas. Everything, therefore, that we have said about introverted thinking is equally true of introverted feeling, only here everything is felt while there it was thought. But the very fact that thoughts can generally be expressed more intelligibly than feelings demands a more than ordinary descriptive or artistic ability before the real wealth of this feeling can even be approximately presented or communicated to the world. If subjective thinking [Ti] can be understood only with difficulty because of unrelatedness, this is true in even higher degree of subjective feeling [Fi]. In order to communicate to others, it has to find an external form not only acceptable to itself, but capable of arousing a parallel feeling in them. Thanks to the relatively great inner (as well as outer) uniformity of human beings, it is actually possible to do this, though the form acceptable to feeling is extraordinarily difficult to find so long as it is still mainly oriented to the fathomless store of primordial images. If, however, feeling is falsified by an egocentric attitude, it at once becomes unsympathetic, because it i then concerned mainly with the ego. It inevitably creates the impression of sentimental self-love, of trying to make itself interesting, and even of morbid self-admiration.
Later, he says of Fi-doms (as OrangeAppled likes to quote):
Jung p389 said:
Their [Fi-doms] outward demeanor is harmonious, inconspicuous, giving an impression of pleasing repose, or of sympathetic response, with no desire to to affect others, to impress, influence, or change them in any way. If this outward aspect is more pronounced, it arouses a suspicion of indifference and coldness, which may actually turn into a disregard for the comfort and well-being of others. [...] With the normal type, however, this only happens when the influence of the object is too strong. The feeling of harmony, therefore, lasts only so long as the object goes its own moderate way and makes no attempt to cross the other's path. There is little effort to respond to the real emotions of the other person; they are more often damped down and rebuffed, or cooled off by native value judgment. Although there is a constant readiness for peaceful and harmonious co-existence, strangers are shown no touch of amiability, no gleam of responsive warmth, but are met with apparent indifference or a repelling coldness. [....]
And a bit later:
Jung p390 said:
Since this type appears cold and reserved, it might seem on a superficial view that such women have no feelings at all. But this would be quite wrong; the truth is, their feelings are intensive rather than extensive. They develop in depth. [...] an intensive sympathy, being shut off from every means of expression, acquires a passionate depth that comprises a whole world of misery and simply gets benumbed. It may perhaps break out in some extravagant form and lead to an astounding act of an almost heroic character, quite unrelated either to the subject herself or to the object that provoked the outburst. To the outside world, or to the blind eyes of the extravert, this intensive sympathy may look like coldness, because usually it does nothing visible, and an extraverted consciousness is unable to believe in invisible forces.
Fi Subjectively
Subjectively, introverted feeling tends to grant an awareness of one's own emotional state, and tends to give continual feedback in real time. Emotions are not themselves to the total of Fi judgments, but serve as valuable input to them. Fi provokes one to continually ask questions like: Is what I'm perceiving good/bad? Is this action consistent with how I feel and who I am? Is this policy consistent with our values as a company? Would I want to be treated that way if it were me?
Fi also tends to lead to continual fine tuning and adjustment. One not only adjusts one's own actions to both one's values and the situation, but one also adjusts ones understanding of others and the world as new input arrives.
Fi principles often tend to emphasize the value of each person and his or her perspective. Typical Fi values often include things like "every person deserves to be treated with respect," and "each person's viewpoint has value." Fi-users tend to value being honest, genuine and authentic over following social convention (this is not to say they are always blunt or direct). This emphasis on being authentic sometimes leads to clashes with Fe values.
Fi-users, like Ti users, can be seen as selfish: both tend to follow internal standards, so externally they may be found inconsistent or lacking. Because both Ti and Fi are highly personalized, they both can be prone to rationalization of selfish behavior (not that Te and Fe are entirely immune to this).
At it best, Fi tends to impart a kind of holistic moral resonance to one's actions and one's life. Life is lived in such a way that it reflects a deep, symbolic understanding of the world and oneself.
It is difficult to detach an Fi-dom's values from the person. In fact, Fi-doms tend to feel personally attacked when their values are attacked or criticized. Therefore, when criticizing an Fi-dom it is important to emphasis one's good intent and to give the Fi-dom time to process the critique before demanding a response.
Fi and Emotional Awareness
Excuse this series of metaphors, but I'm trying to capture what my subjective awareness of emotional state is like as I go through my day.
I feel like my emotions are ever-present companions as I go through my day. In some ways it's like going through one's day with a pet or a child at one's side. My emotions are aware of things I might consciously miss, but I can see their reactions and take them into account. I have a great deal of influence over them, but they are not under my direct conscious control. At times, they become unruly and I momentarily lose control of them. In such cases, I can't help but put everything else aside and attend to them and see to their needs (even when it irritates me to have to do so). They deepen my experience, moment to moment, as though I get to see events through a second pair of eyes. They have wisdom and perception (of a sort), but their insight doesn't always match up with my conscious one. That doesn't mean their insights should be ignored or dismissed.
My control over them is highly influential but indirect. In a way, it is like how I imagine piloting a blimp or dirigible would be. I have controls that allow me a great deal of influence, but external conditions can override my influence and cause me to lose direct control. Even the controls I manipulate directly may cause me to overcompensate and re-correcting is finicky and time consuming. Still, I know them well and they are trustworthy in normal circumstances.