A provider will give you shell access (SSH usually) to their server in exchange for money. You can use it for various things but if you don't know what it is, chances are it won't do you much good to have it.
Most Shell Hosting are used for mIRC to create Ircd - BNC - Eggdrop.
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Shell is a console based environment through sytem which enables you to do variety of action. Its means you can manage your account through the shell. A full unrestricted SSH access to your hosting account allow you to compile your own eggdrops, bncs, and even more.
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It is a network protocol that allows you to exchange data through a secure channel between two ... servers (computers). It is typical for Linux powered server environment. Shell or SSH makes possible for you to send information, passwords, in plaintext and to ensure that you benefit from encryption and integrity of data sent over Internet.
Some web hosting providers give that for the trusted customers only but some promote that as feature included by default to the account so called "shell hosting" (might appear as separated class of web hosting)