A shell account is a personal account on a remote server on the internet. Accounts are often used to run IRC clients, host webpages or use advanced email services.
You often log in via the SSH protocol using some sort of terminal program. When you are logged in, you are presented with a command line interface; a shell. From this shell you can run different programs, or manage your files and settings.
Some shell account providers allow you to have "background processes". This allows you to start a program that will keep running in the background even after you log out from the server. This will allow you to stay connected to IRC 24/7, even when you're alseep and your computer is turned off so you won't miss any messages. It also has many many other benefits and usages ofcourse.
Check this page for some common usage scenarios: UsageScenarios
To sign up for a shell account at Blinkenshell.org, take a look at this page: Signup