The Russians made a whole host of dumb errors, just like we did at TMI. (Similar plants like TMI, such as Perry, continue to operate.)
Chernobyl had some really bad design features including a control rods that INCREASED power output upon initial insertion and a positive temperature coeffient!!!! The disaster, however, was the direct result of human error. A test was ordered to be run (one the Russians were very fond of for some reason) to determine how long the slowing reactor output could sustain the auxillaries such as coolant pumps during a reactor shutdown. To make the test more realistic the emergency core cooling system was disabled!! Outside electrical power was cut off. Next, the reduction of power prior to shutdown was interrupted by the grid operators requesting a temporary INCREASE in output. This added confussion and huge amounts of reserve heat to the reactor. The test was behind so at shutdown they became rushed and careless. Combination of locked out safety systems, poor design, tired & poorly trained operators, end of fuel cycle operating conditions and all that extra stored heat allowed a power run-away. Without emergency cooling system the coolant temperature and pressure ran away and exceeded the burst preasure of the vessel, piping etc. After the explosion most coolant was lost but the nuclear reation was still making a lot of heat - enough to ignite the graphite moderator. Much of the core (>50 tons) was ejected in the initial explosion. Even more was vaporized and released in the smoke plume. A substantial amount melted an flowed out through the bottom of the shattered vessel creating a formation known as the "elephants foot" in the basement. This area remains so highly radioactive that it has been sealed off somewhat and cannot be approached. It continues to give off large amounts of heat. It is likely to do so for many years.