Edinburgh has lots to offer, from being the capital and seat of Scottish government, to being a student city, to the centuries of history present in its city. The history of this city is very hard to avoid when walking around Edinburgh. So, to briefly explain its origins, the name "Edinburgh" derives from the burg, or 'fort', of Eidyn, in Scottish Gaelic, the indigenous language of Scotland which is still spoken today (though mostly in the Highlands). There were settlements in the area as early as the Bronze Age, with the first castle on Castle Rock established by the 11th century. Through Scottish and British history, Edinburgh Castle has been one of the most attacked castles in the world, owing largely to the rocky relationship between the Scots and English, which also account for much of the other historical features of Edinburgh, such as the Flodden Wall or --. Today, the city's entire center - the larger area around the Royal Mile, which is in fact High Street - is a UNESCO World Heritage Site and it holds Scotland's Parliament and the Palace of Holyroodhouse, the King's official residence in Scotland, making it both a vital cultural and governmental center of the UK.
Edinburgh's medieval Old Town is full of winding streets and 'closes', contrasted by the strict city grid and parks of the New Town (remodeled and built in honor of King George III), Edinburgh Castle and the Waverley Train Station nestled right in-between. Newington and Marchmont in the South represent the lively student neighborhoods (and where I lived for 4 years), near the University of Edinburgh's central campus, and border two of the city's best parks, the Meadows (repeating grass meadows used for physical activity and picnicking, as in the name) and Holyrood Park, which includes the dominant Arthur's Seat and Sailsbury Crags, both incredible viewpoints for the city. Those neighborhoods are flanked by Fountainbridge (newer and bustling) and Bruntsfield (more residential) to the West. The West End/Haymarket are in-between Fountainbridge and New Town, an area with another train station, businesses, and oval-shaped parks. Northwest of New Town is Stockbridge, a popular and cute neighborhood famous for it's Sunday market, which also includes the historic and charming Dean Village and the Water of Leith walkway, a wonderful walk along a bubbling stream.
The northern end of New Town becomes Canonmills and leads down to Leith, which is in a way the sister to Edinburgh, before it grew big enough to envelope the former completely, Leith is the port and industrial center of Edinburgh, and site of the Royal Yacht Brittanica, the ship used by the British royal family to travel until the 1990s. Leith also has a popular restaurant scene along its harbor.
If you're a nerd like me, you might enjoy watching this video on this history and map of Edinburgh to better understand its layout!