If I'm getting what you're saying, then you have what is probably a white mounted drive on the desktop, although it may have a firefox icon as its icon (so it doesn't look like a folder or mounted drive).
When you open this, there is a window open with another icon inside for firefox.
If this is the case then what I tell you now will apply to all future applications. Whenever you install an application it will work one of two ways. The easy way is an installer program that will just do it for you, likely asking you to select options as you go. The second way is you download a ".dmg" file, which when doubleclicked creates a mounted drive with the application inside of it.
Take the icon of the application inside the window/folder/drive, and drag it to your applications folder, which is inside your Macintosh HD folder. It will take a second to copy over.
Now, when you want to run the program, go to your applications folder and doubleclick it.
What you called the "Dashboard" is properly called the Dock, if you mean that strip of icons at the bottom of the screen. To keep an application in there all the time, you can either drag the app from your applications folder to the dock (which will leave the original in the applications folder and create an alias on the dock) or you can start the program and once it appears in the dock, you can right click it (or control-click) and select the option "keep in dock" from the menu that pops up.
Hope that covers it.
good luck