Nuclear weapons, or nukes for short, are immensely powerful, one-use weapons in Civilization V. A civilization must have Uranium and complete the Manhattan Project in order to build them. There are two types of nukes: the Atomic Bomb (which requires Nuclear Fission) and the Nuclear Missile (which requires Advanced Ballistics).
Nuclear weapons can be launched from cities or by various units at any tile in range (10 for Atomic Bombs, 12 for Nuclear Missiles). They are truly devastating, especially against field forces, as they are capable of obliterating most units of their time and very seriously hurting the others.
In addition to the damage they do to military units, nuclear weapons leave behind radioactive fallout and reduce the Population of cities in their blast radius, decreasing their productivity and making them easier for infantry units to capture before they can muster a strong defense. Moreover, Nuclear Missiles will utterly destroy cities with a low enough Population.
The Bomb Shelter can reduce the Population loss from nuclear weapons, but practically nothing can be done to defend against them, especially if they are launched from a city. The Atomic Bomb is dropped by a bomber unit, so it is supposed to be possible to shoot it down. Unfortunately, due to a bug that has been present since the game's launch, it is impossible to intercept the planes that drop Atomic Bombs. Since interception isn't an option, your best defense against nukes is to capture any enemy cities that are within firing range of your territory and then sink any enemy naval units that are capable of delivering nukes. Carriers can deliver Atomic Bombs, and Missile Cruisers and Nuclear Submarines can deliver Nuclear Missiles, so if you are at war with a civ that has these weapons, try to target and destroy these units before they can get close enough to your cities to unleash their nuclear arsenal. If you want to stop your opponents from building nukes entirely, you'll either have to cut off their Uranium supplies or pass the Nuclear Non-Proliferation resolution in the World Congress.
AI civilizations that acquire nuclear weapons will start to behave more aggressively toward civs that do not have them. Therefore, you may want to build at least one nuke as a deterrent to nuke-happy AI opponents, even if you don't plan on using it. If you do choose to use nukes, be forewarned that doing so counts as a declaration of war against any civ or city-state whose territory or units are in the blast radius, and be prepared to suffer substantial diplomacy penalties with anyone on the receiving end of your nuclear strikes!