It's possible you didn't drag the associated program into the Applicaitons folder or run the installer, your running the program instead from the window the dmg makes when you open it from the Downloads folder next to your Trash can on the Dock.
Some programs are self contained, everything is one package and you drop it into Applications, others use a installer program to install the program in various locations.
The dmg is a "volume" thus when your shutdown the comptuer it's unmounted, you don't see it, so you go to the site download the program again and it opens itself into a window that your running from again.
However if you look in your Downloads folder, you will see multiple copies of the download file.
Another possible is someone installed Deep Freeze on the Mac, which upon a reboot will set the machne back to what it was before it was frozen, this is security IT measure to prevent people from altering the machine or installing unknown software/malware.
Another remote possibility is you don't have "Anywhere" set for the Gate Keeper in your System Preferences.