To activate user registration on your site, first you need to go to Dashboard => Settings => General and mark checkbox for Membership – Anyone can register.
You can control site authentication from WordPress Customizer => Extas => Authentication. You can control following:
- Site Authentication: On / Off
- BuddyPress Account Activation: On / Off
- Login Popup Heading: Text field
- Login Popup Text: Text field
- Sign Up Button URL: URL field
- Registration Popup Heading: Text field
- Registration Popup Text: Text field
- Reset Password Popup Heading: Text field
- Reset Password Popup Text: Text field
- Captcha on Login: On / Off
- Captcha on Registration: On / Off
- Captcha Type: Image Captcha / Recaptcha
- Social Authentication: On / Off
About WordPres Social Login you can read here.
How to enable account activation via email for users?
In Dashboard => Settings => BuddyPress you need to make sure that checkbox marked for Notifications. Then go to WordPress Customizer => Extras => Authentication and for BuddyPress Account Activation move trigger to ON. In this way after registration on your site users will receive activation emails and they need to activate their account from email. In this case Admin needs to confirm all posts published by Contributors.