It is probably in reference to the Quemoy-Matsu Crisis, the second Taiwan Strait Crisis.

The backstory: after years of civil war, in 1949 the Chinese Communists took over the mainland of China, creating the People's Republic of China (PRC), and forced the nationalists to retreat to the island of Formosa (Taiwan), where they established the Republic of China (ROC). The United States strongly supported the ROC over the PRC, and sought to extend a "nuclear umbrella" to the ROC, openly threatening to attack China with nuclear weapons if they sought to invade the ROC ("nuclear blackmail").

There were, as one might expect, a series of close-calls in the 1950s over whether PRC would invade Taiwan (the various Taiwan Strait Crises).

Quemoy is a small island just off the coast of mainland China, within range of artillery (unlike Formosa). In 1954, the PRC started shelling Quemoy. This put the US in a tough situation because it had already threatened to intervene against the PRC if they attacked Taiwan, but it wasn't clear that a) they actually wanted to go to war over this tiny island, and b) that the use of nuclear weapons to defend this tiny territorial claim (which is, to be fair to the PRC, much closer to their coast than it is to Formosa).

The Eisenhower National Security Council was pretty deeply divided as to what to do. Some thought the US ought to vigorously come to their defense, or otherwise risk a policy of its "umbrella" not meaning much (to both the US's enemies and allies). And there were others who saw no benefit in a war with the PRC (which at best would be a quagmire, and at worst could escalate as the PRC was still a Soviet ally). There was pressure on the US to define which territories it would actually actively defend -- if they were willing to let Quemoy or other similarly-situated islands (like Matzu) be overrun, then that seemed ot be sealing their fate (since the ROC could not hope to fend-off the PRC on its own). On the other hand, if it committed to getting into a big war over tiny bits of territory than it was possibly exposing itself to huge liabilities.

Anyway, the shelling and hostile actions continued through early 1955. There were signs that the PRC might be preparing for a full invasion of Quemoy -- they were constructing airfields, roads, etc. in the areas near the islands. And the US felt it was increasingly staking its reputation on what happened with these islands.

How to deal with this? One approach was to broadcast a nuclear threat, more or less -- Eisenhower had his Secretary of State more or less say that nukes were "on the table," and to argue that they could use them in a purely military way (that would have minimal impact on PRC civilians). This made the NSC very uncomfortable and polled extremely poorly with the American and global public. The American public as well had very little interest in going to war over these islands.

It appears that Eisenhower was completely willing to actually use tactical nuclear weapons if it came to it in 1955. He wasn't interested in actually doing it, if it could be avoided, and his preferred approach would be to deescalate the whole area -- have the PRC and ROC stop messing around with brinksmanship -- but he had made secret plans to defend them by any means necessary. He wasn't particularly worried about the Soviets getting involved (he didn't think they'd want to get involved over Taiwan), but it was a concern of the NSC.

The US increased its own posturing, both in speeches but also, apparently, by flying military planes over PRC territory. They were ready, behind the scenes, to start bombing China both conventionally and with nuclear weapons.

In the end, the US struggled to find good diplomatic solutions (e.g., encouraging the ROC to just withdraw most of their people from Quemoy, but still keep it as a territory, which the ROC would not agree to), but the PRC stopped shelling Quemoy and the whole thing sort of quieted down. The actual, internal PRC reasoning for this isn't quite clear but it appears to have been an effort to appear more responsible to their Asian neighbors. It may have had to do with a growing wedge between the PRC and the USSR that would eventually end up with the Sino-Soviet Split.

The fallout (so to speak) from the situation is that Mao decided it was high time for the PRC to get their own nuclear weapons, so that "nuclear blackmail" could never be applied towards them again. So there's that.

For the full details, see Gordon H. Chang, "To the Nuclear Brink: Eisenhower, Dulles, and the Quemoy-Matsu Crisis," International Security 12, no. 4 (Spring 1988), pp. 96-123.