What is the difference between these concepts, and when should I use one in particular? Does this listing also contain different names for the same general concept?
HTML5 web storage is a generic umbrella term for the new client-side data storage options.
Local Storage is persistent and scoped to the domain. At the moment two flavors are usually mentioned:
Session Storage is non persistent and scoped only to the current window.
Cookies are the old school way of doing all of the above. Stores name/value pairs per domain.