If your account has been threatened, a slim red warning will pop up at the top of your account which reads, "We believe state-sponsored attackers may be attempting to compromise your account or computer. Protect yourself now." Clicking on a link will guide you to a webpage which recommends creating stronger passwords ("with punctuation marks and numbers") and updating your software to the newest version. It should be clear, we're talking not only about hacks but also malware or phishing attacks.
As far as how Google determines when an account has been compromised, the company is frustratingly obtuse, saying "We can’t go into the details without giving away information that would be helpful to these bad actors." We wonder whether the company will tell users when their accountsby the FBI or if the warning system only applies to those living outside the country. Given that the "flame virus," created by the same entity as the Suxtnet virus, was just discovered to have been spying on computers in the Middle East for the last four years, we'd say a warning system against government phishing attacks is a welcome development indeed.