I'll speak on my own behalf of how I experience Ni. It may apply for other NFJs here on the board.
I have a preference to use Ni along with Fe and Ti to solve problems and to make sense of the world around me. It could be just about anything from theory to dealing with people. A lot of my closest friends have said that I read people well. I think it's related to Fe, putting myself in the other person's shoes, their needs and interests, and then making a sudden conclusion about who they are and what they may have experienced. One person I had met, I didn't even know much about them other than their name and common personal interests. But I made a conclusion without effort about the environment that they are drawn to as a symbolic expression of who they are. I could tell in the person's gaze that they were also an introverted personality. I really don't put in a lot of effort to come up with these conclusions. Sometimes I can be off, but most times I am convinced that my intuition is right, and in most cases it has been.
Like what SubtleFighter has stated about Ni being future oriented, it's definitely true. I think I plan things ahead too much, but I am often desiring and designing a plan for the future, which direction I want to go. When I was 12 I knew where I wanted to be when I got into my 30s. I often knew what I wanted and how I was going to get there, and I didn't like it when others stood in my way to get from point A to point B. Ni visions drive me towards where I want to be in my life. A lot of my goals are attached to improving the world, making it a better place, helping others, inspiring other people. I get a feel for what is a person's natural aptitudes and figure out a gameplan to get them to learn the material. I seek to fuse connections between what is being observed to my own inner concept of a theory, for instance.
And, I often find myself drawn to paradoxes in general. I even see myself as a paradox. I'm lucky if I really understand myself completely. When I see things that don't quite mesh with my inner logical concepts of things, I first think, "This doesn't make sense." But then after mulling it over, thinking about the things that a concept has in common with another concept, examining it from different angles, it ends up becoming clearer to me. I often find myself analyzing things to the point I am not even present to them. While details may overwhelm me, I end up providing way too many details to explain my thoughts. I think each angle of a concept is important to truly explain it. I think this may be an example of Ni working with Ti. Overanalysis to the point I'd need a cooling fan in my brain. I want to come down to a conclusion about what I am studying or analyzing. I don't like being left up in the air. "I think this is what the story means. Is it the correct interpretation of the story? Why do I think and feel this way about it? Does this make sense? What course of action should be taken to find out what the story really means?" Similar to what ZBuck posted.