I am using iTunes version 12.5 in Windows 10. I am looking at my Apple Music list of songs and each one has a blue "plus" icon next to it:
When I click on the plus icon, it disappears - what does this do?
The blue cross (which is red, for some reason, on mobile devices) adds the song to your iTunes Library, which is not the same as downloading it. A song being in your Library simply means it's easy to access from any device you're using with your Apple ID. You can download songs in your library to any device, but it may require a separate step. See more here: https://support.apple.com/en-us/HT204839