Nuclear
Nuclear refers to a reaction caused by a nuclear reaction. The main uses of nuclear technology is in nuclear reactors, which use the energy in an atom to create power, or nuclear weapons, which use the energy in warfare.
Theory
The science behind a nuclear device is Einstein's equation 2. Spelled out, mass multiplied by the speed of light squared. This equation is the formula for the energy contained in an atom. This energy is released in a nuclear weapon when a speeding neutron collides with the atom, blasting it apart, and releasing the energy contained.
Weapon
A nuclear weapon is a weapon designed with a nuclear warhead. Nuclear weapons not only produce mass devastation in the blast radius, but also create a nuclear fallout for many years to come. Recently, there has been much fear of a terrorist state or organization getting hold of either a Soviet weapon or creating one of their own.
Trivia
Nuclear weapons are often colloquially called "nukes."