Software is a set of instructions that tell the computer what to do. The greatest hardware in the world will not do you much good if you can't control it.
The operating system, or system software, is the first thing to load when the computer starts. It is different from application software like Word or Firefox in that it controls the entire computer system, including managing what data or programs are loaded into memory, what a user has permissions to do, communication with peripheral devices, and what applications run. Multiple operating systems can be installed on one computer, called a dual-boot system, where the user chooses which OS to load when it starts. You can also run a virtual machine, where you run a special application which lets one OS running inside another OS. You can completely reformat the hard drive and add a different OS than the one your computer came with or start up your computer using an OS that loads from a bootable CD.
Application software must be specifically written for the operating system it will be run on, but some software vendors will release a program for multiple operating systems; for example, you can download versions of Firefox for Windows, Mac, or Linux. Files are collections of data that can be opened and edited using compatible application software. MS Office 2007 can open up files created by old versions of Office, but you have to install a special compatibility pack for old versions of Office to open Office 2007 files.
Most operating systems we use now utilize a Graphical User Interface (GUI) where you can move, resize, open, and close windows for each of the applications you are running. Icons are little pictures that represent your data files and the folders that your files are organized in. There are also text-based operating systems where you have to type in commands to tell the computer to run, edit, or copy a file instead of just being able to click on it with your mouse. These are generally more common on servers that don't need a pretty GUI in order to function. Whether or not a computer has a GUI or a text-based interface, without an operating system, your computer won't accept input, produce output, store data, or process/manipulate data. The hardware and the application software need an operating system in order to talk to each other.