The phrase "reactor core" refers to the center part of a nuclear reactor where nuclear reactions take place. It is like the engine that produces energy in a nuclear power plant. Full definition
When auxiliary batteries were exhausted, the plant was without power to continue cooling reactor cores and spent fuel pools. (scientificamerican.com)
Specifically it covers particle interaction with the metals and alloys used in nuclear reactor cores and hence subject to intense radiation fields. (brightsurf.com)
Without a steady coolant supply, a hot reactor core will continuously boil off the water surrounding it until the fuel is no longer immersed. (scientificamerican.com)