Simplified so smart children can understand it:
Nuclear power is just using radiation to heat water (or something else that is later used to heat water), then using the steam to turn a turbine, connected to magnets creating electricity in wires near the magnets. To do this we need a source of radiation that is consistent. To much and it gets so hot it melts everything near it ( 'melt down'). Too little and the radiation is not sufficient to keep going.
Radiation is when you spew out atomic particles - we will be focusing on spewing out neutrons. When they hit things it can heat them up and/or cause other atomic particles to spew out of the thing it hit. A chain reaction is when a material has enough radiation that they spew out enough particles to continue the process indefinitely.
Cold criticality is the point where you have enough radiation to create a chain reaction but the heat being generated is not that much. Not enough to make steam to turn a turbine. No electricity yet, but you are on the right track. Also, this is safe as it won't get hot enough to melt the machinery.
Delayed criticality when the chain reaction is strong enough to make steam to turn a turbine but not enough radiation to worry about. Things are delayed enough for you to control the situation. The neutrons are are going strong, but not fast enough to worry about a melt down.
Prompt criticality is when you get a run away reaction that keeps getting hotter and hotter. This is scary. Because it causes a melt down. This is unlikely to create a nuclear explosion because unless you intended to build a nuclear bomb, something melts and everything fizzles out.
Super criticality is what you build everything well enough to so that it won't melt down. This is called a nuclear explosion. Luckily you have to really work hard to build things this tough.