I was an adult when the Harry Potter books came out. I was aware of the hype but, for some reason, it never occurred to me to read any of them. Anyway, I was babysitting my nephew one night (I was staying over) and had forgotten to bring a book with me. The Harry Potter books were sitting on my nephew's bookshelf so I picked up the first one and got stuck in. I was instantly hooked and went out and bought a set of the first three books. I subsequently bought the next three as individual books and bought the last one on the day it was released. I loved the first four books but not so much the last three which are much darker. Also, Order of the Phoenix is way too long.
I loved them because, as previous posters said, they really do have everything in them. A mistreated orphan, a boarding school (very familiar fiction to me as I grew up with the Malory Towers, St. Clare's and Chalet School books), an incredibly colourful magical world that somehow co-exists with the more mundane muggle world and a truly evil character hell bent on world domination. And so many themes - friendship, family (or lack of ), good vs. evil, loss/bereavement. Also, so many vividly drawn characters - major and minor. The three main characters experience lots of adventures and we bond with the characters as we go through those adventures with them. They also transform from children into adults in the process, so we are witness to that transformation. There's a lot of humour in the books too. For example, Neville encountering a boggart in the form of Snape and picturing him in his grandmother's clothes. Or Ron's attempt at asking Fleur to the ball, knowing he was out of his depth.
My teenage DD isn't a big reader, despite my best efforts. She watched the movies one Christmas and went on to read the books. She's a major HP fan (but perhaps inspired more by the movies than the books) and we would have endless discussions about the characters and the plots. One day I was exasperated by the endless questions and discussions and said, "You do know it's not real, you know". DD said very quietly, "Yes, but I like to think it is". And I felt like the worst mother in the world 
. Anyway, I think her response encapsulated the appeal of the books. Who wouldn't want to receive an invitation to a magical boarding school (delivered by an owl, no less) where they can learn how to become wizards and fly on broomsticks and encounter all kinds of magical creatures and have all sorts of exciting adventures?