Nuclear power is power (generally electrical) produced from controlled (i.e., non-explosive) nuclear fission reactions. Electric utility reactors heat water to produce steam, which is then used to generate electricity. In 2007, 14% of the world's electricity came from nuclear power, despite concerns about safety and radioactive waste management. More than 150 naval vessels using nuclear propulsion have been built.
In a Zombie Apocalypse scenario (or any kind of Apocalypse, for that matter), the inability to maintain the nuclear materials therein pose a considerable hazard for all life within distance of relevant prevailing winds of reactors in both power plants, and naval vessels.
Once a nuclear isotope has undergone fission, the core material will continuously produce both heat and radiation for a time and intensity directly proportionate to the material's radioactive half-life. In the modern day, this is carefully monitored with a variety of failsafes.
A real life example of a narrowly avoided meltdown caused a failed cooling system was the Fukushima Daiichi nuclear disaster of 2011, though this was only the case because the reactor suffered heavy damage from a simultaneous earthquake and taunami; it's a testament to their safety that the incident was as minor as it was, requiring multiple natural disasters in the same location and at the same time to even create the conditions necessary for a failure. In a zombie apocalypse, the extensive maintenance a nuclear reactor requires will inevitably fall short as the economy, roads, and transit systems stagnate and decay. Thankfully, the main way a reactor automatically shuts down in an emergency (dropping the control rods all the way) results in an inert reactor that takes power and people to restart, so an apocalypse shouldn't cause too much trouble.
There are over a hundred nuclear power plants in the United States, and hundreds more nuclear waste storage facilities that likewise require constant refrigeration. There are also many nuclear power plants around the world.