I was there. In the (Swinging) Sixties when Sloane Ranger was born. Eleventh door along the King's Road from Sloane Square. Would love to explain it all to you but Wikipedia does that really well already. Also I feel a list coming on!
I've heard (admittedly this was secondhand and many years ago) they are the young, beatiful, very trendy and expensive women who live in the neighborhood (or at least are seen there) of Sloane Square in London.
Actually, I wonder if the idea is similar to the people who live on the Upper West Side in New York...
Edit: I think Diana was one before she got married. I wouldn't call her one after that though.