As far as I can tell, they were never sent through the post (if you know of any examples, please comment). They were just hand-delivered notes containing informal invitations, short apologies, brief questions, little flirtations, and so on. In the 20th century, their function was taken over by the phone call, and in the 21st century, by text messaging.
I had no idea of the long history behind notes like these. Via Alan Jacobs most recent always excellent newsletter.