A nuclear weapon (also called atomic or thermonuclear weapon) is an explosive device which employs either nuclear fission or nuclear fusion in order to create a destructive blast and/or emit high doses of lethal radiation.
The Vulcan species developed the technology early in their warfarring history but rather then engage in a widespread disarmament program, they instead developed lesser forms of the "dirty weapons" such as neutron weapons designed to kill organic matter but leave the resources as well as structures behind. (TOS novel: Spock's World)
Earth first developed nuclear weaponry in 1945 when the United States of America, under a program codenamed the Manhattan Project, created and detonated a device in Trinity, New Mexico. Two more nuclear bombs were then used against Japan to expedite the ending of World War II. Though the United States had only built two bombs by this point, their intention was to create the illusion that they had many more in its arsenal. (SCE eBook: Lost Time)
Nuclear weapons were also extensively used during World War III.
In 2155, the Romulans used nuclear weapons against Earth's NX-class starship Discovery, leading to its destruction. Romulan vessels continued to carry such weapons into the 23rd century. (ENT novel: Beneath the Raptor's Wing, TOS episode: "Balance of Terror")
In 2266, Spock had an atomic missile from the USS Enterprise loaded with certain chemicals and fired into the core of the planet Metamorpha in order to stabilize it. (TOS comic: "The Peril of Planet Quick Change")