Most programs you download end in .DMG, which stands for Disk Image.
When you double-click a DMG file, it mounts the disk image in Finder, the same way it mounts USB drives when you plug them in, or mounts a CD/DVD if you stick on in (if your Mac has a CD/DVD drive).
For 99% of the applications out there, you're supposed to then drag the application from the mounted DMG, and drop it in your Applications folder. Then you eject the DMG, and you can delete it the DMG if you want, because you're copied the app to your Mac.