Great explanations, guys! I can’t really think of much to add.
One aspect of Chernobyl’s terrible design is that the rate of reaction actually increased with increasing temperature (referred to as a positive temperature coefficient of reactivity), which made it more likely that a runaway power spike could occur.
Chernobyl also had a positive void coefficient of reactivity, so that when the coolant flashed to steam, the rate of reaction increased.
The pressurized water naval reactors I qualified on had negative temperature coefficients of reactivity and negative void coefficients, which contributed to their passive safety features. Even if the pressure vessel cracked open and all of the coolant was lost, because of the loss of neutron moderation, the nuclear reaction would at least halt. (Of course, you would then have to deal with the decay heat.)
This was not the case with Chernobyl, which continued to increase its rate of reaction until it blew itself apart, dispersing the fuel and halting the fission reaction.
More detail here:
Another contributing factor to the accident was the lack of training of the plant operators, who were grossly deficient in their knowledge of nuclear reactor physics and engineering.