Edinburgh offers an embarrassment of riches in the way of attractions so I have picked a good cross-section of things that are historical, fun, indoors, outdoors, natural, man-made, some you pay for, some are free, and all are appropriate for the whole family.
If you only have one day and you are travelling with youngsters my top 3 would be Edinburgh Castle, Camera Obscura & Mary Kings Close all at the top end of The Royal Mile but offering history, fun and drama.
If you only have one day and you have no youngsters in your party then start with the Open Top Bus which will give you a good tour of the city and an idea of where you want to go next time whilst dropping you practically at Edinburgh Castle, the centre of everything followed by delving into The Museum of Scotland . Geographically an easy walk from one to the other and close to some great bars and restaurants.
If you have a few days then you will find a lot of these attractions are all based round The Royal Mile so a day or two there and then the outposts such as The Royal Yacht Britannia in Leith, The Modern Art Gallery or The National Air Museum are a more an effort to get to but well worth it.
If the weather is good or you are geared up for some good walking then I have included numerous options including Arthur's Seat, Calton Hill and The Water of Leith. But hopefully a scroll through all the things I recommend will inspire you to go and try something new.