To me, a DMG file is roughly the equivalent to a ZIP file in Windows.
They're both containers.
You don't usually care that the container is called .DMG or .ZIP, you usually just care about what's inside of the container.
So on that note, you don't "install" a DMG. It's just a container, so you open it, and then do something with the contents (like "install" it).
If the content of the DMG is an application that you want to install on your Mac, you usually just drag 'n drop the application over to your Applications folder. The application is then "installed". At that point, the DMG isn't needed (unless you wanted to keep a copy of the original download), and you can just eject it and delete it.