Edinburgh is a city not lacking in places to stop in and drink, and in my time here, I've tried a fair few. But the first one I always recommend to friends coming across the pond (or even just across the continent) is Finnegan's Wake. It's an Irish pub between the Royal Mile and the Grassmarket on a windy street with a handful of good pub choices.
The atmosphere is excellent, with live music every night and a generally full house. There aren't many tables to sit, so you get a lot of folk standing/half dancing on the floor, but it's good fun and the cover bands are, for the most part, really good. The seating they do have is interesting, with a bit of privacy. The bar staff isn't the fastest in Edinburgh, but they pour a beautiful pint. I would recommend this as a pub not to be missed.