String cUrl
You'll want this to match exactly what you see when you're looking at your community's Administration Settings. Don't add any extra trailing slashes!
String startUrl
This should be the URL relative to the value you put for cUrl, including the leading forward-slash ('/'). For example, if you want to land the user at the Home tab, startUrl should be set to "/home/home.jsp" (again, note the inclusion of the leading forward-slash).
Think about it this way: If you concatenate cUrl and startUrl, exactly as written, you should end up with a valid URL to the desired page or resource within your community.
String developerName
This is the DeveloperName field from the AuthProvider object. Or, you can find the value by looking at the URL Suffix field on the Auth Provider detail page.
You can find this value dynamically by querying the AuthProvider object.