This might be about Jane Seymour, the third wife of King Henry VIII. She was one of the few wives not executed, but died at childbirth while bearing his only son. Another possibility is that it's about Jane Ormsby-Gore, a British woman Mick Jagger was involved with.
Brian Jones, who was The Stones guitarist until his death in 1969, played the dulcimer, an instrument you play on your lap by plucking or strumming the strings. Jones could learn just about any instrument very quickly. He had just recently learned how to play it when they recorded this.
Keith Richards: "Brian was getting into dulcimer then because he dug Richard Farina. We were also listening to a lot of Appalachian music then too. To me, Lady Jane is very Elizabethan. There are a few places in England where people still speak that way, Chaucer English."
Mick Jagger: "Lady Jane is a complete sort of very weird song. I don't really know what that's all about myself. All the names are historical but it was really unconscious that they should fit together from the same period." (thanks, Bertrand - Paris, France, for above 2)
Jack Nitzsche played the harpsichord, which gave this an Elizabethan feel. Nitzsche was a prolific keyboard player and producer. He died in 2000 at 63.
This was left off the US version of Aftermath. It was on the Flowers compilation.
This was the basis for the Neil Young song "Borrowed Tune," which appears on his Tonight's The Night album. He sings the lyric, "I'm singin' this borrowed tune I took from the Rolling Stones." (thanks, John - UK, England)
Chip Monck, who handled lighting and production duties for the Stones in the late '60s and early '70s, often played an instrumental version of this song over the sound system after the band left the stage. He said it was "like a madrigal, really. Have a good evening, get home safely, we look forward to seeing you the next time around."