A nudge is any persuasion attempt that doesn't put someone on the defensive. If you've spent any time in the business world, you've probably arrived at this technique on your own without ever hearing about it. The nudge is important because it represents the understanding that for the most part other people won't listen to you.
People will listen to you only for as long as politeness dictates, and the nudge is the admission that whatever gentle prodding you can slip in during that time is pretty much the best you're gonna get.
Andy Grove of Intel came up with the term, even though he uses it in a context specific to management: the nudge is persuasion a manager uses with associates that falls short of a direct order. Note that often he nudges his direct reports instead of issuing commands. Why bother? Why not bark orders like a drill sergeant? Well, for a few reasons. He might want to give the report leeway to override him. He might also be playing a hunch. But primarily, the nudge seems to be about maintaining social harmony.
A nudge is a way to spread influence without incurring the cost of resentment. It's invariant with respect to social disharmony, so as long as you keep your nudging light, you can nudge and prod and never have to worry about the costs.
Another way to think of a nudge is that it's anything short of argument. Argument is a powerful mode of discussion and discovery, but you almost never get to have one, because in our world usually what an argument does is close minds.
The only people you can count on to hear an argument are people who are stuck with you. Family members, business partners—that's about it, and most of them don't want to. You can count on good friends to hear an argument, but not on the typical friend, because the typical friendship doesn't extend past "having a grand old time". Imagine then how willing a stranger is to hear you argue.
If you must argue with someone, being respectful helps, but no amount of respect can turn an argument into a non-argument. People recognize when they must contest with information, and that's the part they don't like.
Because arguments are so good at uncovering the truth, if you never argue, then you can be sure to arrive at something phony eventually. That's why one of the best qualities a relationship can have is to allow for the occasional "penalty-free" argument now and then. Then you get the best of both worlds: authenticity without the resentment.