Toolbar Notifications allow you to view and moderate comments right from the toolbar, anywhere on your site or across WordPress.com.
To use notifications, a user of your site must go to the Jetpack admin page and link their account to a WordPress.com account. That ensures that they have their own channel for notifications. This step isn’t necessary if you’re the main user who connected the Jetpack site to WordPress.com, that serves as the channel for your user.
Once you’re connected (main user) or linked (other users) from the Jetpack admin page, click on the notifications icon in the toolbar and log into WordPress.com if you’re not already. Now you’re set to view and moderate comments on your site and from anywhere on WordPress.com. Note that in order to moderate comments from the notifications toolbar menu, a user must have comment moderation privileges (if the user can moderate comments elsewhere, they will be able to in notifications also).
For more details on using Notifications (and keyboard shortcuts) see the WordPress.com support page.