Volume & Interruptions
With Android 5.1 Lollipop we still don't have a truly "silent" option for notifications. You can only go to vibrate only mode, and no completely silent option. However, with Android 5.0 Google changed this entire process and introduced controls to only let priority notifications come through, or simply none. Not even alarms. This wasn't a good way to handle it, so Google's fixed this in Android 5.1 Lollipop.
None, Priority, and All now have icons next to the text, to make things more clear for users. Priority can now be more closely controlled where you can set a time where all notifications will come back on, and the "None" option can be fine-tuned to stop all notifications until a previously set alarm goes off, then notifications turn back on. This makes it easy to turn off everything at night, or while at work, but the system could still be handled much better.
In settings > Sound & notifications > Interruptions there's now a "downtime" option that allows you even more control over the None option in sound settings. Fumble around and get familiar with it. It isn't the best, but better than it was with Android 5.0 where we had no true silent mode unless we turned off EVERYTHING.