In the coming weeks, you may begin to see real-time notifications while you're using Twitter.
As you can see in Twitter's example graphic here, Christina is DM-ing Ducker: "Hey Ducker!"
Ducker gets to watch – either in horror or sheer glee – as this pops up in real time, distracting his attention from the very important things that are happening on Twitter at that exact moment.
According to Twitter's blog post announcing the feature:
When you’re logged in on twitter.com, you will receive notifications if someone has replied, favorited or retweeted one of your Tweets. You can also receive notifications for direct messages and new followers. They’re fully interactive, so that you can reply, favorite, retweet, and follow right from the notification.
You'll be able to turn this off on a granular level under in the Web Notifications menu under Settings, but note that the Web Notifications menu won't be present until this feature rolls out for you. You may have to be on high alert for weeks -- weeks!
Once the feature is live for you, you can opt to be notified when your tweets are retweeted, your tweets are marked as favorites, you're mentioned in a tweet, you get a new follower or you're sent a direct message. Leave some of them on, all of them on or turn all of them off. It's up to you, Ducker.