As above really. But I think that one thing to mention is that home premium lacks automatic backup to a network drive which you would expect it to include.
Do all the computers need to have pro for this or? Brought home a laptop from work which is running pro as it wasn't installing enterprise from a USB stick for some reason. Anyway, couldn't get it to work to my home premium desktopRemote Desktop I find very useful which is Professional minimum.
Pretty much as others have said.
Home Premium at a minimum.
Professional if you need to use virtual machines, want the backup functionality or use more than 16GB of memory.
Ultimate if you want the bitlocker security stuff.
Remote Desktop I find very useful which is Professional minimum.