This is the upper part of a memorial pillar on Bristo Square, with a beautiful cast-iron lantern on top. The bottom part represents the City of Edinburgh coat of arms (you'll see it better when you click on the image to get a larger picture).
The coat of arms consists of a shield with a three-towered castle on a rock, with two 'supporters': a maiden and a doe. The maiden was probably adopted as a supporter because Edinburgh Castle was once known as ‘Castrum Puellarum’ (Castle of the Maidens). This Latin name was probably given after a Gaelic or Brythonic name sounding like ‘maiden’ but meaning something else. The doe or hind represents St Giles, the city’s patron saint.
Finally, the crest above the shield takes the shape of an anchor and rope probably because the Lord Provost of the city also held the office of Admiral of the Forth.