This one is by no means difficult, but I often forget about it. I see no reason why Firefox should remember what I download, and it bugs me to see a huge list of previously downloaded files whenever I open the Firefox Download window.
Disabling this feature is simple. Just go to Tools -> Options, and switch to the Privacy tab.
Un-check the box next to Remember what I’ve downloaded. Problem solved.
While we’re on the subject, you can also suppress the Downloads window from showing at all when you start a new download. Just switch back to the Main tab and un-check the box next to Show the Downloads window when downloading a file.