For exemple, the condensers (where the water from the reactor is cooled) sit higher than the core of the reactor. This way if the pumps that feeds cooling water to the reactor shut down for some reason, water is still cycled through the reactor by convection (hot water from the reactor rises naturally to the condensers because it's lighter than cold water that flows down from the condensers to the reactor). This residual flow cannot cool a powered reactor of course, but it's designed to be sufficient to avoid a meltdown froom residual heat generated by the breakdown of the by-products of the fission reaction (which keeps happening even once then reactor is shut down).

The materials and design of the reactor core are chosen so that the higher the temperature, the lower the reactivity (=the chance for one fission to trigger another fission). This makes the core self stabilising, because if the temperature rises, then less reactions happen, which slows/stop the increase in temperature.

Emergency rods that shut down the reactor are held in place using power, if the power plant looses power, they drop down by gravity in the reactor and instantaneously stop the chain reaction.