My experience with Ni is similar to what tovlo and Seraphim have expressed. I think of it like a computer process that's always running in the background, internalizing as much information as possible while putting it all together based on my specific understanding of reality at any given point. Basically, Ni dominants are constantly taking in unfiltered elements of the whole picture and attempting to comprehend this data in a manner that makes sense to us. However, this isn't necessarily a conscious process (it's not for me most of the time, at least).
That's why many Ni dominants have the so-called "aha! moments." In certain instances, Ni decides (after thoroughly examining the information it's amassed on a subconscious level) to share information with the conscious mind that may seem new, spontaneous, or out of nowhere to the Ni dominant. As an INFJ who relies strongly on my tertiary process, my Ti process then critically evaluates this information based on its own system of logic (which may or may not match external systems of logic). Fe also looks at the info to see how it will affect others around me. These additional processes occur almost instantaneously and in conjunction with Ni, and are just as intrinsic to overall functioning, so it's hard for me to think of them as extraneous when talking about aha! moments.
When intuition is at work in me, it is rarely accompanied by any specific feelings. It's just like knowing something that you've already known for a long time on some level. I tend to be able to trace my way back through the web of information to the source of the knowledge, but I have to make an effort to do so. I trust what comes to me through intuition and don't generally second guess it. If someone asks me to step them through the entire thought process (1-2-3) I will do my best, but this can be exhausting because explaining it also requires an explanation of the complex conceptual frameworks that we utilize (unique to each Ni dominant). I believe this is why some of us have a tough time with sensors.
Many psychics now refer to themselves as "intuitives." Some even feel there's no distinction between intuition and psychic awareness (i.e., the strict psychological definition of intuition) and think that anyone can develop this kind of intuition. Recently, I've been working with this idea and allowing my intuition to play a bigger role in my life (letting Ti relax a bit), and have been noticing major changes in myself and the way I see the world around me. I'm feeling more connected, my thoughts and actions more natural, and I'm substantially more in tune with others around me.
Thanks for starting this thread, Jungian Trip. Very interesting to hear peoples' thoughts/feelings on this.