Do Not Disturb (DND) is not actually a new feature but did gain some functionality in the iOS 7 update.
Previously with DND, you could enable it and not receive notifications while your phone was locked. Two exceptions: repeated calls (2 calls from the same number within 3 minutes) and calls from your favorite contacts. If however your phone was unlocked, DND would allow calls and notifications.
In the updated DND, you can still receive calls from your favorite contacts and repeated calls, but now you have the option to silence all notifications while your phone is unlocked. So if you were working on a paper in Pages with DND enabled, you would not be alerted of messages, calls or other push notifications. There is an exception to this: If you are in an app while you receive a push notification, you will hear sound and see the visual notification. For example, you are texting person A and person B sends you a message, you will now get a visual and audible alert. Another good example would be Tweetbot: If you are scrolling through your timeline and you get a DM or an @reply/mention, you now will hear and see the alert.
In the settings, you can choose whether you want DND to be always silent or only when phone is locked. You are still able to have scheduled DND time which is great if you don't want your phone going off during the night. Do Not Disturb is a handy feature to have when you want to use your device and not be bothered by calls, emails or anything else.