Not a long back Google Webmasters tool started showing the health of websites on the webmasters' home page. All sites that are verified or unverified are shown with their thumbnails and a brief messages about any recent critical issues. This helps greatly to those who owns a couple of domains. They need not to check every individual site for its health.
By default all websites are shorted by their health. Thus, sites with health problems are shown at the top of the list. There is also an option to view sites alphabetically. If Google thinks that there are some issues with your site's health, it would show an icon with "Check site health" link in front of the website. The icon simply indicates that the site health has been compromised. To see the detailed report click on the "Check site health" link.
Site Health Parameters
Site's health is determined by considering three parameters. Some more can be added in future but at present only these three are participating in the judgement.
If your website is found to be infected with malware, then it will raise the health alarm.
Critical health issue may be shown to you if Google finds that you have removed any important page of the website by using URL removal tool.
If some of important pages are blocked from crawling in robots.txt file.
You might be thinking of how Google decides if it is an "important page"? Well, even this is not totally perfect, but at present Google only counts the number of clicks a page was getting before its removal or blocking. Higher the counting, more is its importance. However some other signals may be added in future for this purpose.
After getting the warning, it is your turn to work on those issues. After fixing them just wait for some days. It may take about 4-5 days for the warning to disappear from your Webmaster Tools account.
I forgot to tell that if you have more than 100 websites in Google Webmaster, you would not see any alert on the home page, rather site health information are available individually on the top of Dashboard for each website. But if you have 100 or fewer sites, all these will work for you.