Type about:config into your address bar and press enter.
In the search box, search for browser.download.dir and reset it so that it points to your desktop
Then search for browser.download.lastDir and also reset it to your desktop
If this fixes the problem then it is likely that a file in the directory it was pointing to before was corrupted. This happened on my old hard drive, which was an indication of the drive slowly failing. That may not be the cause in your circumstance, just something to think about though.