Shadow Copies is the informal term for Volume Shadow Copy Service (VSS), also known as "Previous Versions." All three names point to the same feature: the automatic saving of copies of files. The purpose of the service is to allow you to restore these backups of files that you have lost for any reason—file corruption, for example. VSS works in conjunction with both System Restore and Windows Backup. If it's properly set up, simply right-click a file name from within Windows Explorer and choose Restore Previous Versions to access this feature. Vista—or Windows 7, for those already using the beta of Microsoft's new OS—will open the Properties dialog for the selected file or folder, and from there you'll select the previous version you want to restore. Some users won't have to set up VSS, but in case your system isn't already configured to save versions of files.
More info: Setup