A "reactor core" is the part of a nuclear power plant that produces energy. It consists of the nuclear fuel that undergoes moderated fission, the moderator, the first order coolant, and the containment vessel for them, . In some nuclear reactors these parts can be the same thing: in some reactors, water serves as both a coolant and a reaction moderator.
Although the term "reactor core" might sound nefarious, its a pretty simple concept. The "reactor core" is the part that produces energy in the form of heat. The rest of the reactor facility turns that heat into electricity through a turbine, as well as offering further cooling, containment, and waste storage. The same terminology could be supplied to a car: it has a "core" where combustion takes place and then a variety of parts that utilize and transform that power.