The new feature offers story recommendations "based on what you've searched for and clicked on" in the past. Make sure your Google Personalized Search is active.
This page has more info about personalized news search.
Here's a look at what I found after spending a small amount of time on the site.
In the left column, you'll find a "most popular" link. I'm assuming this lists the most popular stories clicked on by Google News users for a given period of time. I can't find any documentation as to how often this page is updated but I'll check with Google on Monday.
Yahoo News has offered a most popular, most recommended and most emailed feature for years. From what I can tell, at least as of now, only one "popular" list is available from Google while Yahoo offers "popular" lists broken down into various news categories. Both services offer these lists via syndication.
The other new feature from Google is called "Recommended for XXX (Your Gmail address)" and can be found directly below the Top Stories on the Google News home page. On my page, three stories (the max) were recommended.
This help page explains how some of this works.
By signing in to personalized news and keeping Personalized Search enabled, you allow Google to track and save your news selections. Then, Google News can automatically recommend relevant stories just for you by using smart algorithms that analyze your selections. The algorithms compare your tastes to the aggregate tastes of other groups of similar Google News users. Simply put, we recommend news stories to you that have been read by many other users who've also read similar stories as you in the past.
In addition to the three recommended stories listed on the Google News home page, you should also see a new selection in the far left column labeled "Recommended." In my situation tonight, I found about 20 recommended stories listed. Of course, it would seem to make sense that the number of recommendations would be related to how much you've clicked and read using Google News, the Google Personalized being active, and the amount of content that Google believes is relevant.
I checked a few (not all) of Google's country sites that offer news like Google News Canada, Google News New Zealand and Google News UK and noticed the recommendation and popular story features were available. I also check Google News for a few non-English speaking countries and did not notice these features. For example, Google News Argentina and Google News Italy.
As to the quality and usefulness of the recommendations, it's going to take some time to determine what Google suggests is useful and worthy of a click. I'm going to give it a few weeks.
Of course, Google is by no means the first service to recommend news stories it thinks might be of interest. I've blogged about and used Findory since it first launched and have found many stories I would have otherwised missed without it. Findory also recommends material from blogs and can even function as an RSS aggregator that will also recommend stories you might find interesting. I'm still hoping that Greg Linden (Findory's CEO) and his team develop a mobile version of the service.
If you're not seeing any recommended headlines, select and click on a few stories and then refresh the page. Google News is only going to show you recommended headlines when it thinks it has some for you.