You might receive one of these when you are getting dumped by your erstwhile partner. I don't think it is a coincidence that the a recent movie called Dear John includes a Dear John letter as one of its plot points.
According to Wikipedia:
While the exact origins of the phrase are unknown, it is commonly believed to have been coined by Americans during World War II. Large numbers of American troops were stationed overseas for many months or years, and as time passed many of their wives or girlfriends decided to begin a relationship with a new man rather than wait for their old one to return.
As letters to servicemen from wives or girlfriends back home would typically contain affectionate language (such as "Dear Johnny", "My dearest John", or simply "Darling"), a serviceman receiving a note beginning with a curt "Dear John" would instantly be aware of the letter's purpose.
A writer in the Democrat and Chronicle of Rochester, NY, summed it up in August 1945:
"Dear John," the letter began. "I have found someone else whom I think the world of. I think the only way out is for us to get a divorce," it said. They usually began like that, those letters that told of infidelity on the part of the wives of servicemen... The men called them "Dear Johns".
Here's another source that concurs:
Early in Anthony Trollope's 1864 novel "Can you Forgive Her?" the heroine, Alice Vavasor, writes a letter that is central to the novel's plot development. She breaks off her (second) engagement by writing a letter, addressing her fiancee "Dear John" and then repeating this same greeting several times throughout her letter. So I imagine this very popular London novel was the origin of the concept, and that the idiom was further popularised in the USA during the second World War as suggested.
Here is more information about "Dear John" from Wikipedia:
"A Dear John Letter", or "Dear John" is the name of a popular country music song. It was popularized by Ferlin Husky and Jean Shepard, and was a crossover country-pop hit in 1953.
The song played on the concept of a Dear John letter while referencing the United States' involvement in the Korean War. Here, a young woman, whose boyfriend John was stationed in Korea, writes stating that she is breaking off the relationship. To add to the heartbreak: She is marrying John's brother, Don, and wants her photograph back (because — according to the lyrics — Don wants it now).
If you want to know more about the song lyrics here is the link:http://www.lyricsmode.com/lyrics/f/ferlin_husky/a_dear_john_letter_lyrics.html