Do I need to keep .dmg files?

No, you do not need to keep the .dmg file. I presume you've already double-clicked the file and followed through whatever installation process was presented to you(?). If so, the application will be installed - probably in your Applications folder - so you can safely delete the .dmg file.

.dmg files are disk images. A good explanation of them can be found at http://en.wikipedia.org/wiki/.dmg

20" iMac Intel Core Duo 2GHz 1GB RAM Mac OS X (10.4.8) 30GB iPod with Video

I'm kinda obsessive/compulsive about some things, so I usually keep the .dmg files on my desktop until I've used the applications a few times, just to make sure I don't have to reinstall, re-download, etc.

But in all the years I've done this, I've gotta say I've never once had a use for the .dmg files again, so it's probably best just to trash 'em right after you've installed the application on your machine.

I'm actually having the same problem. But, whenever I try to delete the .dmg file, the Trash says that the application is already in use. Whenever I manage to delete that too, the application won't work.

You need to EJECT the mounted disk first that mounts when you double-click the .dmg file. Drag the mounted disk to the trash to eject it. You should then be able to trash the .dmg file.

You say the application doesn't work when you manage to trash the .dmg file. Have you actually gone through the application's installation process prior to trashing the .dmg file as mentioned in my previous post above?

20" iMac Intel Core Duo 2GHz 1GB RAM Mac OS X (10.4.8) 30GB iPod with Video

You don't need the .dmg files after you've installed.

I've got in to the habit of keeping the dmg files safe and back them up occasionally just in case I ever need to reinstall stuff.

I have gone through the installation already. At least, I think I did. The program in question is Firefox.

Well, after installation I tend to keep the .dmg file in the same folder where the application was installed. Then if the application is ever corrupted I can just delete it, and do a reinstallation from the .dmg without having to go back and download it again. And yes, I have had an application get corrupted to where it wouldn't run and used the .dmg again.