In journalism, a nut graf puts the story in context and tells readers why the story matters. Many editors would say the nut graf is the most important section of a story, since it tells the readers why they should be reading the story.
As a journalist you would either hear your editor refer to your nut graf or see the word written out on one of your stories after it had been edited. You would not, however, see the term in a proper sentence.
For more on how to craft a good nut graf, check out this article on writing news stories.