Commercial nuclear fission reactors are operated in the otherwise self-extinguishing prompt subcritical state. Certain fission products decay over seconds to minutes, producing additional delayed neutrons crucial to sustaining criticality. An example is bromine-87 with a half-life of about a minute.  Operating in this delayed critical state, power changes slowly enough to permit human and automatic control. Analogous to fire dampers varying the movement of wood embers towards new fuel, control rods are moved as the nuclear fuel burns up over time.
In a nuclear power reactor, the main sources of radioactivity are fission products along with actinides and activation products.  Fission products are most of the radioactivity for the first several hundred years, while actinides dominate roughly 10 to 10 years after fuel use.
Most fission products are retained near their points of production.  They are important to reactor operation not only because some contribute delayed neutrons useful for reactor control, but some are neutron poisons that inhibit the nuclear reaction.  Buildup of neutron poisons is a key to how long a given fuel element can be kept in the reactor. Fission product decay also generates heat that continues even after the reactor has been shut down and fission stopped.  This decay heat requires removal after shutdown; loss of this cooling damaged the reactors at Three Mile Island and Fukushima.
If the fuel cladding around the fuel develops holes, fission products can leak into the primary coolant. Depending on the chemistry, they may settle within the reactor core or travel through the coolant system and chemistry control systems are provided to remove them. In a well-designed power reactor running under normal conditions, coolant radioactivity is very low.
The isotope responsible for most of the gamma exposure in fuel reprocessing plants (and the Chernobyl site in 2005) is caesium-137. Iodine-129 is a major radioactive isotope released from reprocessing plants. In nuclear reactors both caesium-137 and strontium-90 are found in locations away from the fuel because they're formed by the beta decay of noble gases (xenon-137, with a 3.8-minute half-life, and krypton-90, with a 32-second half-life) which enable them to be deposited away from the fuel, e.g. on control rods.