Literally, a "nudge" is a small or gentle push. It urges someones to move in a particular direction, but doesn't force them.
Richard Thaler and Cass Sunstein are behavioral economists who wrote an influential book entitled "Nudge: Improving Decisions about Health, Wealth, and Happiness." It concerns public policy. The idea is to encourage desired behavior through gentle "nudges" rather than force.
For example, in the United States, many employers offer retirement savings plans called "401(k) plans." They are intended to encourage workers to save for their own retirement, by offering a tax advantage and, often, financial contributions from employers. They are a good deal, but workers tend not to bother signing up for them, perhaps because they don't want to have contributions taken out of their paycheck.
An example of a "nudge" is a rule that says that employers will automatically have 3% of their salary taken out of their paycheck and put into their 401(k) account _unless they file paperwork saying not to do it._ It doesn't restrict their freedom. They don't need to do anything difficult to stop the contribution, they just need to ask. But by making a 3% contribution the default, it sends the message that "this is a good idea," and it makes it slightly easier to do it than not to do it. It doesn't force employees to contribute, it just "nudges" them.
The concept has become popular. As a result, if we see the word "nudge" in a book about behavioral economics or public policy, we assume it means "Oh, one of Thaler and Sunstein's 'nudges.'"