[1] The term was coined by Saul Kripke in his 1970 lecture series at Princeton University, later published as the book Naming and Necessity.
[2] As an example, consider the phrase "The 43rd President of the United States of America": while the 43rd President of the United States is actually George W. Bush, things might have been different.
Bush might have lost the election, meaning that the 43rd President might have been Al Gore or Ralph Nader instead.
"The 43rd President of the United States of America" is thus a non-rigid designator, picking out George W. Bush in some possible worlds, Al Gore in others, and yet other people in other worlds.
Non-rigid designators are defined by contrast with Kripke's notion of a rigid designator, which picks out the same thing uniquely in every possible world; while there are possible worlds in which the 43rd President of the United States is Al Gore instead of George W. Bush, there are no possible worlds where George W. Bush is anyone other than the man who, in fact, he is.