In the mouths of everyone else, a postscript (from post scriptum, a Latin expression meaning "after writing" and abbreviated P.S.) is a sentence, paragraph, or occasionally many paragraphs added, often hastily and incidentally, after the signature of a letter or (sometimes) the main body of an essay or book. In a book or essay, a more carefully-composed addition is called an afterword, e.g., for a second edition, an afterword, not usually called a postscript, is written in response to critical remarks on the first edition. The word has, poetically, been used to refer to any sort of addendum to some main work, even if not attached to a main work, as in Soren Kierkegaard's book titled Concluding Unscientific Postscript.
In the age of e-mail, postscripts have become unnecessary: if one has something to add to a letter, one simply edits the e-mail before sending, while if one has something to add after a letter has been sent, an entirely new e-mail must be created.
Perhaps the most common postscript found in love letters is "P.S. I love you!" This title was given to at least two popular songs, one by Rosemary Clooney and one by The Beatles.
P.P.S. is a "Post-postscript" and allows the letter writer to add even more thoughts after the first postscript. To continue, a third postscript would be a P.P.P.S. and a fifth would be the P.P.P.P.P.S.
Cf. other publishing terms: afterword; appendix; addendum.