Notifications (a.k.a. OSD) is one of those UI features which seems fancy at first, but quickly gets in your face. Possibly originated on the Mac, the annoyance has now spread to Firefox, Fedora and Ubuntu as well. Here’s how to quickly switch them off:

Firefox

Enter about:config in the location bar. Search for this preference, and double click.
browser.download.manager.showAlertOnComplete

Fedora and Ubuntu

sudo mv /usr/share/dbus-1/services/org.freedesktop.Notifications.service /usr/share/dbus-1/services/org.freedesktop.Notifications.service.disabled