Here you can choose how users in this institution will authenticate to the site. Institutions should have at least one method of authentication tied to them.
The internal authentication plugin checks the site database for usernames and passwords. Users have to have an account for this to work, which can be done by allowing registration via the registration form, or using the Add user or Add users by CSV pages.
The none authentication method allows anyone in. Users can pick any username and password. This should only be used for testing!
The other authentication methods generally authenticate against an external source, such as an LDAP server or remote application. They have their own settings, which you can configure in a pop-up window when you add them. Some methods are disabled because their required PHP extensions are missing or the outgoing connection to the external sources cannot be established. For example, the Persona authentication plugin needs the cURL PHP extension and the HTTPS connection to verifier.login.persona.org.
One setting they normally share is whether accounts can be automatically created on the site when a user signs in for the first time (as opposed to accounts being created previously by the administrator, which users can then log in to).