Shell access on a Unix-type server is access to a command prompt on the server itself. This enables you to issue commands as a user of the system - for example, for changing permissions on files, copying, moving, deleting and such. To access a shell account from your home machine, you can use WinSCP - for Windows like explorer GUI or use the excellent Putty if you are running Windows.
Shell access is useful even on a shared account, especially for installing scripts and more complex file changes which are difficult or impossible via an FTP client.
If you want to learn about shell accounts and the Unix-type command line, you can try some of these guides:
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