Notifications let you know if you have an incoming or missed audio or video call, IM, or anything else when you're not actively using the Lync app. Notifications also let you know if there's an issue with the app, or if you need to take action. The Lync live tile will also show a number that indicates any missed notifications.
If you don't turn on Push notifications you may miss incoming audio or video calls, IMs, and other alerts.
To turn on Push notifications do this:
From the my info screen, tap More (…) icon, tap settings, scroll down to the Push notifications section, and then tap On.