I work at a publication at my college and one of my best friends (and the main reason I am still a part of this publication) is an INFJ. Being an introverted intuitive in an organization dominated by extroverts and having to intimately work with an unbalanced ESTP is not easy. When we were in the production room together, her doing actual work and me just milling about hanging out, she related her stress to me and said
"It's so comforting not having someone constantly looking over my shoulder at my computer screen."
While she is saying this I am literally standing over her shoulder looking at her computer screen. However, she intuitively knows that I am simply blankly staring at her screen, or at the very least not judging her.
Another instant is where I had parted ways with her and two other ladies for the night, but events then led me to go in the same direction as them. Therefore, I decided to try out my skills and give these oblivious females a scare, so I quickly and stealthily ran through them. The two others had little panic attacks, one of which had a screaming fit, the other claiming she nearly peed her pants, while my INFJ friend, un-phased, said "I knew you were coming."
The later case was a premonition, but the previous case illustrates the innate ability to pick up on vibes. With my intuition being extroverted and auxiliary, I cannot simply do this. I must observe the situation and then come to the same conclusion.
Another way of looking at Ni, and possibly easier, is to observe it at the other end of the spectrum, as the inferior function.
The aforementioned unbalanced ESTP is a prime example of that. He has many times overstepped his bounds and it was once even necessary, after he had violated four significant protocols in one week, to have a harsh reprimanding of him during an open meeting. However, he just doesn't get it. He thinks he will be re-elected next semester, but nobody trusts or likes him. He is unable to pick up on these vibes and has made a very inaccurate prediction as to his future. This is what happens when one is out-of-touch with their Ni.
Another similar example is of my ESXP cousin. He has been dating an ESFJ for four years and has been living with her for about two years. Before thanksgiving, she left him because of his non-commitment. While he may be a good man and not unbalanced, he was still completely oblivious to her emotions, and completely surprised at her actions, whereas everyone else in the family, though not fully expecting it, was not surprised.
Si is where one takes in sensory data and lays it out onto a sequential grid. Ni is where the sensory data is taken in, but instead of being put into a grid, they are thrown into a pot where connections are made within. While this may seem obvious, it is essentially Ne within the mind. My Ne stays outside my being and connects the dots as they already are, all the data is left outside. Ni brings it within and does the work inside the head. This is also why Ni is a judging function and Ne is a perceiving function. Being that the data is brought within the body and processed within the body, Ni is less malleable because everything is set. With Ne, data is processed outside the being, and determinations made are all the more fluid because the data remains outside the body.
That was a lot longer than I meant to be