Notifications help let you know what's going on in the wiki that affect you, by sending you updates through the Echo extension. When someone sends you a new message on your talk page or thanks you for your edit for example, you'll see a notification which you can view by clicking the icon that has a small badge and a number. This number you see represents how many new notifications you've received.
You can view notifications by either using the popup menu, or going directly to Special:Notifications. This special page lets you see much more and provides wider control.
These notifications can also be sent to two places, which is by email and/or on web. For example, you may want to only have some updates be sent through the web interface for things that are lower-priority, and higher-priority updates through email. They can also be sent to both.
By default, you'll be sent notifications from other wikis. For example, if someone posted a new message on lego.shoutwiki.com at your talk page and you were on another wiki, you'd be still be able to see this update. If you'd like to opt-in in opt-out for certain notifications and control how they're sent, you can do so by going to your preferences.