An update to the Blizzard launcher and associated tools this afternoon has been causing problems for folks running various versions of Windows. The bug has a chance of occurring when you get stuck at a "Waiting for files to close..." dialog box when updating. Blizzard support staff was quick to announce a fix, which is pretty easy to do.
To solve this, go into the task manager (you can get to that by right-clicking on the bottom taskbar on your screen and selecting "Start Task Manager," or by pressing the control, shift and escape keys all at once) and end any and all of the following processes in the "Processes" tab:
The full Blizzard statement after the break.