An “Operating System” is the system that all other programs “Operate in”. An iPhone, iPod Touch or iPad is really a portable computer. An “App” is really a computer program.
The operating system on a PC is “Windows”. The operating system on a Mac is “Mac OS X”. The operating system is what you see after you turn on a computer for the first time, but before you install any programs.
The OS in iOS stands for Operating System. Thus iOS is the base software that allows all other apps to run on an iPhone, iPod Touch or iPad.
Some operating systems include games or utilities. For example Windows comes with Solitaire and iOS devices come with programs to play music and videos.