When you download Firefox, you're actually downloading a disk image (dmg file). When you double click the disk image, it mounts the disk, as if there were a "real" drive attached to your computer. In the attached disk, you will see the Firefox icon. To "install" it, simply drag it to your desktop or appropriate folder. Copied.
Or, if you also see an Applications folder in the drive next to the Firefox icon, you can also drag the FF icon onto that Applications folder. It is actually an alias (shortcut) of your actual Applications folder on your hard drive. After you make the copy, you can unmount/eject the disk image and throw out dmg file.
Now you REALLY have Firefox.