In my opinion, this is the best "plain English" explanation of Ni, that I've ever seen. Credit to the forum member Harley.
In my own words, I would say Ni is a lens of perspective.
It's about looking inwards and reflecting on how something is the product of of someone(s) past actions, looking at what it is today, or contemplate on what it might mean in the future. For example an Ne might look at a measuring cup and think "hmm, sure this can used to measure ingredients, but if I want to get creative I could use it as a hat, a holder to hold my pens, a vase to hold my flowers etc." Ne wants to find ways to use a measuring cup outside the conventional way of measuring food ingredients.
Ni on the other hand, would look at that same measuring cup and think, "this measuring cup is used to measure food ingredients, which in the past was the domain of woman since they were expected to carry on domestic duties. Yet today both men and women are expected to take share in such duties, so now this measuring cup is not a symbol of oppression but rather equality. Also notice how this measuring cup is only using the metric system even though I am not American. The measuring cup can also be a symbol of American capitalism and dominance, since it uses a measuring system no other country is using". Ni is not interested in manipulating the cup in any way like Ne, but rather sees how the cup is the product and symbol of different thoughts and movements (feminism, capitalism etc.)
As for examples, I think a really good example of Ni would be the domino scene for V for Vendetta
If you look at the first comment in the link for that video, someone writes that the dominoes represent the people that were killed during the regime, and the last domino represents V himself. The dominoes aren't just dominoes they are symbols that represent the oppression of the government. And notice how the dominoes are arrange so that they create a V? This is meant to represent that all those people in died during the regime, their deaths, and their pain, is what made V. He is a man who is made up of the pain and suffering of others.
I can't "define" Si, but, based on my observations of Si users, I'll list some traits, that I relate to it. Si users seem to have the most keen awareness of their physical environments, out of all the types. For instance, when my ESTJ sister visits, she considers my room to not be clean, because she's quick to notice, that I didn't dust my fan and dresser. My ESTJ father, used to try and impress me, as a child, with his ability to walk up to a clover patch, and pick out a four leaf clover, from a standing distance, within five seconds. A child I know of, who would seem to have good use of Si, could pick out anything, that stood out in the environment. For instance, at the ballpark, he noticed, that one of the light fixtures, was turned in the opposite direction of all the others. His ESFP mother, said she never would have noticed.
This honed sensory awareness probably develops, because it is minute sensory details, that trigger Si's disengaged mode. For instance, seeing a pair of earrings, that draws up a flashback type memory. There's no conscious way of telling what sensory cue will trigger this, so they just become very in tune with their whole sensory environment, as it's always some minute sensory detail, that triggers it. It remains subjective, fitting the qualifications for an introverted function, because the pair of earrings would have to have subjective importance, to trigger that flashback memory.
Si, also, seems to be a bit "rigid", much like Te. It isn't questioning. It has sort of a "What is, is, and what works, works." vibe. If you want to correct an SJ child, there's no need to tell him why his behavior was bad, only to tell him, that it was bad. He'll commit it to that storehouse, and recall, when he runs into that situation again, that he shouldn't repeat his behavior. There's no need to know why he shouldn't. If an SJ runs into a situation, that is familiar or relatable to them, they just use what they know has worked before. In a situation, that is completely foreign, they are apt to be very hesitant and uncertain of how to proceed.
Si catches a lot of negative flack, but any Te user would have to be a bit envious, of the extra bit of efficiency, that Si allows the xSTJ. The wider range of experiences the Si user has, the narrower his focus is allowed to become. An xSTJ, with a few years on the job, will kick ass, at his/her job.