East of Edinburgh Castle, the historic castle esplanade becomes the street known as the Royal Mile, leading from the castle down through Old Town to the Palace of Holyroodhouse. The Mile, as it's called, is actually made up of one thoroughfare that bears, in consecutive sequence, different names—Castlehill, Lawnmarket, Parliament Square, High Street, and Canongate. This thoroughfare, and the streets and passages that wind off it on both sides, really were Edinburgh until the 18th century saw expansions to the south and north. Everybody lived here: the richer folk on the lower floors of houses; the less well-to-do families on the middle floors; and the poor highest up.
Time and progress (of a sort) have swept away some of the narrow closes and tall tenements of the Old Town, but enough survive for you to be able to imagine the original profile of Scotland's capital. There are many guided tours of the area, or you can simply stroll around at your leisure. The latter is often a better choice in summer, when tourists pack the area and large guided groups have trouble making their way through the crowds.