Can you explain why some apps get installed, and some apps appear as disk drives?

I don't even have the right language to explain this. The vast majority of applications get installed where LaunchPad can find them. But I have one app that LaunchPad doesn't see. This appears as something that looks like a disk drive on my desktop. I can click on the disk drive icon, and there's the app, and I can run it from there. Hopefully that description makes sense.

The main difference I saw when I installed it, was that MacOS told me it was from an untrusted source. I figured out how to run it anyway. It's KeyPassX, a well known key vault, so it's safe. It's approved by my company for installation. It just didn't come from the App Store.

So, why does it appear as disk drive on my desktop? Can I get it to work through LaunchPad like everything else? I've seen other apps behave this way, but I always deleted them because I didn't want anything that wasn't normal on this machine. But I really need KeePassX, so I'd like it to run the way other apps run. Make sense?

Maybe I just need to copy them to the applications folder? I keep thinking about this as if it were a Windows machine where installs have deep roots and you cannot simply copy apps from one location to another.

There are multiple ways to install apps on a Mac. Some apps just come ready to run, all you need to do is to copy them to the Applications folder and you are done. That is what most apps from the Mac App Store do, at least that is my experience. Other apps require installation of some sort. They show up when you get them with an icon that looks like a wooden crate. You click on the crate to run the installer, and the installer puts things where they need to be, ending up with the application in the Applications folder. Finally, some apps come distributed in files that end in .dmg. That indicates that the installer is on a virtual disk image (hence "dmg") and that to install it you click on the drive icon to mount the image as a drive, and then in that virtual drive you will find either the app itself, to be dragged to the Applications folder, or an installer app that works exactly the same as the installer mentioned before, or it may be the crate again. I don't use Launchpad, but all three installation methods should end up with the app in Launchpad as well, AFAIK.

Hope that helps some. macOS typically has multiple ways to get to the same end.

To follow up on Jake's excellent post, the OP's description sounds like the app is being run from the .dmg and has not been installed.