A nut graf (or nut
graph) is the paragraph or sentence that explains the point of a
journalistic piece. Nut grafs are also used as transitions between the lead and the rest of the story, and relates the current story to past relevent events. It is also often called the
gist graph.
When used in hard news articles, the nut graf is almost always the same as the lead. Since hard news stories are most often very direct and use simple language to communicate the news, lead sentences describe the point of the story and generally do not need much expansion.
The first sentences of soft news stories - or even hard news stories that use soft leads - are generally not as quick to the point, so they need a secondary lead (the nut graf) to expand on the point and relate it to the rest of the story.
Some soft news stories deal with current issues but they start out by profiling one person or incident. The nut graf ties the individual or specific incident in to the issue.
The following example illustrates how a nut graf can expand upon a descriptive lead.
Soft lead: Safe schools are important, say civil libertarians, but so are schools that don't infringe on the privacy of their students.
Nut graf: The stabbing death of a teenager outside of his school last week has led to questions about the effectiveness and ethical dilemma of surveillence cameras in schools.
Timeliness is one of the key factors that determine what is and isn't news. Nut grafs can also explain why the piece is timely by making reference to other recent related issues, as shown in the above example.
While there are several names for the nut graf, it is most commonly called this because this paragraph contains the "nut" or the "gist" of the story.
Resource:
http://www.worldmag.com/world/olasky/truthc8.html
http://www.poynter.org/dg.lts/id.52/aid.34457/column.htm
The sample lede and nut graf were from an assignment that I wrote.