Premise

The premise of a work is the fundamental concept that drives the plot.

Most premises can be expressed very simply, and many films can be identified simply from a short sentence describing the premise. For example: A lonely boy is befriended by an alien; A small town is terrorized by a shark; A small boy sees dead people. Basically, it's the way you'd pitch the script to a Hollywood Producer.

The premise is not the plot itself, but a description of the core of the setting, characters, genre, and/or conflicts which drive the plot. It doesn't summarize every aspect of the story, just the core conflicts around which the story is based.

For example, the classic "a knight has to save a princess from a dragon" story, cliché as may be, is just an expression of a basic plot, Save the Princess -- and the core plot of Man versus Beast.

Nor is the premise the entirety of the story. The premise of Harry Potter can be summed up as "a boy discovers he's a wizard and an accidental hero in a hidden wizarding world". The bulk of the story is devoted to this plot and the consequences thereof, but there are many, many other plotlines which accrue over the course of seven books.

The premise is highly related to a few tropes:
 * Willing Suspension of Disbelief is the continuous negotiation between the storyteller and the audience that convinces the audience to accept the premise and its consequences, regardless of their deviations from "reality".
 * The "Unicorn In The Garden" Rule is the idea that the premise should account for -- either directly or indirectly -- all divergences from reality necessary for the plot to progress, and that the author should not introduce new premises without somehow tying them back to the original in some way. It also urges that the most fantastic elements of the story should be stated or implied by the premise early in the story (i.e. the first chapter, first episode), so that readers aren't surprised by Ass Pulls later on.
 * Genre Shift and Cerebus Syndrome are additions of new premises into existing stories, typically done at the beginning of a new Story Arc.

A few common premise types:
 * Audience-Alienating Premise -- the show is well-done, but the premise scares away potential viewers
 * Idiot Premise -- characters never manage to resolve the easily-solvable premise
 * Recycled Premise
 * X Meets Y -- Describing the premise in terms of two other (probably more popular) works.

Literally every story in existence has a premise, even the Shaggy Dog Stories, so we don't need to post examples here except to explain the concept.