The first thing to know about Edinburgh is that most Americans pronounce it wrong. It is not meant to have a harsh "o" sound at the end, but rather to be pronounced "Edinburra". Even if it sounds a bit ridiculous to you at first the effort will be appreciated.
Of all of the cities I've been too (not that there are too many) Edinburgh is one of my favorites. It has a certain old-world charm that really can't be rivaled. It's quintessential British nature is just stunning and the way that all of it looks up at the Castle is beautiful. Part of its charm is that it has never become too painfully touristy. Sure there is the Royal Mile and a few particularly touristy shops and restaurants, but even those are kept incredibly tasteful and never too packed. You can visit the Elephant House, "the birthplace of Harry Potter", see the crown jewels, and hit Jenners, a department store nearing 200 years old, just to name a few. I've been a few times now, and have never had a bad experience. In my opinion it's one of the most underestimated tourist destinations in Europe. Few Americans I know have been, but all who have love it.
I was lucky enough to get the chance to go last Saturday for an art history field trip to the National Gallery. It's an amazing gallery so if you're into art, and even if you're not I would highly recommend it. It's completely free of charge so there's no reason not to give it a try. It houses a wonderful collection of art from around Europe and also some of Scotland's greatest works. This coming Friday I'll be back again to see Swan Lake so I'll let you all about that too!