Concierge features email notifications, allowing you to send an email on specific user events. Each email can be enabled in Concierge's settings, with the content of each email customisable via the Craft System Messages utility.
When a user registers on your site, you can nominate a user group to be the "moderator" of the registered user. Concierge will send an email to each member in the nominated user group, letting them know someone has registered on your site.
Coupled with the Craft Settings → Users → Settings → Deactivate users by default setting, this is useful for letting moderators know they need to moderate new user registrations.
If your users are deactivated by default, you'll likely want to let them know they they are activated. Concierge can send the user an email letting them know their user account has been activated.