A virtual machine is basically a computer without hardware. It exists purely as a program.
Instead of taking a physical harddrive and plugging it into a physical machine, you create a virtual harddrive (just a big file on your actual harddrive). Instead of inserting a CD into a physical drive, you choose an iso image containing a CDs data in the VM settings. You can specify the amount of RAM and some settings like how the network connection works or whether it uses USB devices.
When you start the VM, it does the same things a computer does: look for a (virtual) DVD or (virtual) HD to boot from.
You can now install an operating system on this virtual machine just like on any physical computer (and you must do this, because a VM simulates just the hardware; without an OS it just sits there, like an actual computer would).
Where a virtual machine goes on the hard drive: the software that simulates the computer is an installed program like Photoshop or Word. Then there's a configuration file storing all the specifics about the computer (just a small text file, not even a megabyte.) Finally you have the virtual hard drive, which is just a file. Usually you'd use a dynamically sized virtual hard drive, meaning the VM will think its quite big (say you set it to 200GB), but the actual size it takes up on your hard drive depends on how full it is. If you're only using 15 of the 200GB, it's only 15GB in size.
If you download something while in the machine, it ends up on the virtual hard drive. If you want to exchange files between host (your PC) and guest (the VM), you can use a shared folder. The computers are connected to each other as if they were in the same LAN or wireless network.
VMWare provides big companies with commercial VM software.
VirtualPC is a Microsoft program primarily geared towards running say Win98 inside Win7.
VirtualBox is a free allrounder, I use it all the time to test new versions of linux or Win8.
VMWare can even run MacOS, I have a virtual SnowLeopard machine running inside my Win7.