When you download any addon you need to extract it into a temporary folder. Then you copy the folder within that folder into the addons folder.
If you look at your Interface\AddOns folder you should see a folder called nUI. There should be nother other characters in that folder name. Can you check and see if it is showing something else ? Also, in the nUI folder should be a file called nUI.toc. If that isn't the case that would explain what has gone wrong.