Manager role
The default manager role enables users assigned the role to access courses and modify them.
The Manager role is a 'real role', similar to Administrator (but much safer to use).
The way permission checks work in the Moodle code is that there is a function called has_capability. For admins, has_capability will always return true, no matter how the roles are set up.
However, the Manager role is a normal role, like Course Creator, or Teacher. By default the Manager role has almost every capability but, because it is a normal role, you can edit that role if you choose (there is no way to edit what permissions an Administrator has). Best-practice might suggest that Admins should normally use a Manager role, and not use an Administrator account.
This is similar to the way you are recommended not to log into Linux as root.
The Manager role therefore allows a site Administrator to give very powerful roles to others who are assigned a Manager role, but without having to give them a full Administrator role.
Assigning the role of Manager Sitewide
You can give someone the Manager role sitewide(to enable them for instance to add new users) by going to Settings>Site Administration>Users>Permissions>Assign system roles, selecting the Manager role and moving over your chosen user.
When you do so, users in that role will have access to only some of the items in Site administration. They do not have access to areas such as Security, Server, Plugins, Advanced Features, which are reserved for those in the Site administrators group. They have access to most of the tools for User, Course and Grade system settings and tools.
Specifically the site-wide Manager role can see these in Site administration:
Users
Accounts
Browse list of users
Bulk user actions
Add a new user
Cohorts
Upload users
Upload user pictures
Permissions
Define roles
Assign system role
Check system permissions
Capability overview
Courses
Add/edit courses
Backups
General backup defaults
Automated backup setup
Grades
General settings
Grade category settings
Grade item settings
Scales
Letters
Report settings
Language
Language customisation
Front page
Front page settings
Front page roles
Front page filters
Front page backup
Front page restore
Front page questions
Reports
Comments
Backups
Course overview
Logs
Live logs
Question instances
Security overview
Statistics