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. Ultimately, those failsafes rely on its nation's economy and transportation system - either to bring the workers to the power plant, or to transport the coolant, fuel, and maintenance parts that keep the cooling system running. A real life example of a narrowly avoided meltdown caused a failed cooling system was the Fukushima Daiichi nuclear disaster of 2011. 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. But if the failsafes fail, an all too real possibility, the results can be catastrophic.
After a reactor's control systems fail, in a mere matter of days, the highly toxic fuel rods (most of the radioactivity comes from fission products, and not the actual uranium/plutonium) will boil away the water that they are being held in, causing them to heat up even more and cause a radioactive fire. This meltdown is not as destructive as a nuclear bomb (where the fission or fusion reaction unleashes megatons of force), but the radioactive material in the atmosphere will kill many survivors, domesticated animals, wildlife, and plants. The heat causes a large radioactive fire which rages on for days, and even after the embers burn out, the material will continue to emit radiation into the air and water of its environment, contaminating the area for hundreds, perhaps even thousands of years to come. As the winds blow this debris and cloud matter miles away, it will become an invisible killer of many survivors.
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.