Several major websites have offered push notifications with Chrome (and other supported browsers) for some time, including sites like Facebook and eBay. Today Google+ is now supporting this feature, enabling users to receive notifications even when the site is not open.
You can enable it by opening your Google+ settings, scrolling down to the Notifications section, and enabling "Get Google+ push notifications on your web browser." That's it, you will begin to receive said notifications as long as Chrome is running (including Chrome running in the background). This is what they look like, for reference:
Comments on the source G+ post indicate some users are having trouble receiving all their notifications. Two separate people on the AP team tagged me in posts so I could take a screenshot, and while the notification appeared when I checked manually, I did not receive a push alert in Chrome. While I was in the middle of writing this post, I received the above pop-up.
If you also run into problems, make sure you have notifications enabled for the categories you want from the G+ settings.