Famous Last Words are defining last words of a character to the point that they represent the very climax of that character's development, leaving a lasting impression on other characters and fans.
This is difficult to write despite normally being just a few words. Any misuse and it can come off as cheesy. Done perfectly and it can even garner the respect of fans that hated a certain character. An affectionate subversion is that characters utter what they expect will be their last words but survive after all.
Compare Sedgwick Speech where a character gets killed after their speech or one-liner, usually in an ironic fashion. See also "Facing the Bullets" One-Liner (where a character says something before knowingly facing death), Pre-Mortem Catchphrase (when a Catchphrase ends up being their last words) and Dying Curse (when the words are a curse). See also Dying Declaration of Love and Dying Declaration of Hate. Often a Tear Jerker.
Contrast Lame Last Words which are innocuous or pathetic — after all, not everybody has the right to famous last words. However, such last words can actually become famous if memetic enough.
Also compare and contrast Ironic Last Words where a character says something that turns out to be ironic in a certain fashion.
It should be noted that this doesn't exclusively refer to the saying for when someone makes a statement that’s soon (fatally) proven to be wrong.
Massive Spoilers Within, so don't go further if you don't want them.
