Basically a core, a collection of fuel assemblies, generates heat. The heat is removed by coolant. In light water reactors (LWRs), the water coolant both moderates neutrons (from fast energies of fission down to thermal energies in thermal equilibrium with the moderator (coolant) and fuel). In boiling water reactors BWRs), the coolant boils in the core, and the steam is collected above the core and passed through steam lines to a turbine. A set of turbines, one high pressure and two or three low pressure turbines, drive a generator, which produces the electricity (through induction). In a pressurized water reactor (PWR, or Russian VVER), the coolant circulates in a primary coolant loop, and the heat is transferred from the core to a steam generator. The steam is produced on the secondary side of the steam generator, thus keeping the primary coolant and secondary coolant separate. The is passed to a set of turbines, which drive a generator to produce electricity.
BWRs are primarily controlled with control rods during operation. The control rod positions are adjusted periodically, as are the patterns of control rods, in order to distribute the power and burnup distribution in the core. PWRs use soluble boron (boric acid) in the primary coolant. The dissolved boric acid is buffered by lithium hydroxide (LiOH). Control rods are generally not used in PWRs, although some designs use special types of 'grey' rod to adjust power distribution in the core.
CANDU reactors are a type of pressurized water reactor that uses heavy water (D2O), which doesn't absorb neutrons as readily as light water (H2O), so they can use natural uranium. Otherwise, they function the same way as LWRs of the PWR type.
Other special reactors include liquid metal (cooled) fast reactors (LMFRs), gas-cooled reactors (GCRs), and molten salt reactors (MSRs).
In LMFRs liquid metal like sodium or NaK, or even Pb-Bi, are used to remove heat from the core. The liquid metal does not moderate neutrons like water does, so the neutron energy spectrum remains fast, in keV to MeV range. The liquid metal then transport the heat to a heat exchanger, where the heat is transferred to another liquid metal loop, or perhaps directly to a water loop, which feeds turbine.
GCRs may be fast or thermal depending on the amount of moderation in the core. For gas reactors, graphite is a typical moderator. One can either use a Brayton power cycle to drive a turbine, or the gas is circulated through heat exchangers to produce steam in much the same way steam generators are used in PWRs.
The simplest answer is that the power generation is done by using turbines, which is the same as most conventional power plants.
The nuclear fuel is simply the source of energy used to create hot steam which then spins the turbines.