I am new to Mac and I am trying to set up a Mac Mini as a dedicated music server. when I download J River it appears to work OK but when I shut it down the program disappears from the desktop as if it was not saved.
On a Mac, you typically do not install applications. They are distributed as .dmg files, which are disk images.
These are mounted to the system as a virtual drive (i.e. shows up on the desktop and the Finder sidebar) and you then drag the application file from there to your Applications folder to "install" it.
It sounds like you are running it from the disk image rather than the Applications folder.