First and foremost, Edinburgh is widely regarded as one of the most beautiful cities in Europe, and is a UNESCO World Heritage Site. It's small enough to quickly get comfortable, but big enough to keep you interested and entertained through the seasons. You can walk to just about anywhere you need to get to, but the transit system is fantastic as well. The cost of living is pretty reasonable, at least as far as capital cities go.
The city centre is split in half: Old Town and New Town, each with their own distinct flavour of nightlife. Old Town is a twisting, winding tight grid of beautiful stone buildings and great clubs, pubs, bars, and restaurants. The largest university in town – the University of Edinburgh – lies in Old Town, and the area is crawling with students, kebabs, coffeehouses and cheap pints. New Town (an 18th century addition) is a bit more posh, and the nightlife crowd tends to be a bit older. This is where you'll find all your high street shops and expensive cocktail bars. Speaking of which, if you're after a good cocktail, head to Bramble Bar, rated one of the top 20 bars in the world, on Queen Street.
The food in Edinburgh is as varied as its inhabitants. There are great noodle bars (Yocoko among the best), legendary Indian places (Mother India's), and of course haggis burgers aplenty. Teviot Row House is the University of Edinburgh's student union, and is the oldest student union in the world. The old library within has been converted to a bar, but the old décor remains, and you can enjoy a £2.50 pint whilst sitting next to books which are older than your grandparents.
As any young resident of Edinburgh will know, Cowgate is the centre of Old Town nightlife, with great spots like Sneaky Pete's, Cabaret Voltaire, and The Bongo Club. The night scene in New Town is a bit more spread out, but there are plenty of gems to be discovered, like The Stand Comedy Club, and Panda & Sons.
Perhaps what Edinburgh is most known for internationally is the Fringe Festival, which takes place yearly for the entire month of August. The Fringe is the world's largest art festival, and features over 3,000 shows from 51 countries in over 300 venues. During the month of August, the population of Edinburgh literally doubles, from ~500,000 to a million, and for a few weeks the city becomes the most vibrant, creative, and fun place in the world.
Any student in Edinburgh will invariably learn to love the rain, the hippies fire-dancing on Calton Hill, the taste of Buckfast, the natural beauty of Scotland in general, and the overwhelmingly-friendly and international population of the city itself.