As I recall, the royal burgh of Edinburgh was based on the Royal Mile (from the Castle to Holyrood) and the southside. A new town (Princes Street and the big posh hooses on the northside) was begun in the 17th century.
North and southside refer to the Nor Loch which was the big loch and marshy ground, roughly where Princes Street Gardens and Waverley Station are today.
The Royal Mile is built on the tail of the volcanic crag that formed the base of the castle. North Bridge was built to connect the old and new towns and span the gap between the crag and the flatter ground of the new town.
The bridge was completed in the 19th century and is about 1125 feet long and 70 feet tall, at the highest point.
So yes, the bridge was built to span the ravine between the Royal Mile and Princes Street. Oh, and water has always run through Edinburgh. In fact the Scots invented water.