Standard Troubleshooting procedure:

Goto Tools Menu | Select Options.
In the new window, select the “Main” tab at the top. Make sure that “Show the downloads window when downloading a file” is unchecked.
In the same options window, select the “Privacy” tab. Make sure that “Remember what I’ve downloaded” is checked.

In some cases, your download history file can become corrupt. In this case you need to close firefox. Open your firefox profile folder.

Delete downloads.rdf. Restart firefox. (your downloads.rdf file will be recreated)