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. On a dedicated server, you will probably also have "root" access (like Administrator access in Windows) and you will be able to compile and install programs, update and patch your system and a whole lot more.
To access a shell account from your home machine, you can use SSH if you are running Linux, or use the excellent Putty [chiark.greenend.org.uk] 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: