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First, the design of the RBMK-1000 reactor was faulty. By having a void coefficient that varied greatly with reactor power the design was inherently unstable. Second, the operators were able to override the safety systems. At the time of the accident practically all the control rods were removed from the core. Third, the accident occurred during a test of the turbine which was poorly conceived and carried out. Fourth, the emergency scram rods which would shut down the reactor took 20 seconds to operate and their design was such as to drastically increase local reactivity in the core, compounding the original problem. Fifth, the operators and engineers were very poorly trained and in any other country would have at no time been on the job. Sixth, the safety culture of the plant at the Russian plant was non-existent. The plant did not even have a containment building . A peculiar feature of the RBMK-1000 is a graphite moderator. During the disaster this caught on fire and is responsible for the more widespread distribution of radioactivity that occurred. In all North American reactors the moderator is water...not prone to catching on fire!
These and several other reasons all combined in the unlikely series of events that led to the disaster. A paper I co-wrote in University is presented here and goes into much more detail including a timeline of events (this version was edited from the original class to be an engineering ethics term paper). Also see an excellent web site here.
That Chernobyl had a massive effect on people is beyond debate. Many people point to the devastation shown on the "Kid of Speed" website as a condemnation of Nuclear Power. In general the social and emotional impact is incalculable, the economic consequences are huge, the health impacts are *apparently* widespread. The death toll is only at 48 however (31 immediate, 14 radiation, 3 Cancer) and 1800 Cancer cases above expected have occurred. Significant certainly but compared to the human consequence of, say, Bhopal, Chenobyl is not an exeptional disaster.
Let's not let the human factor get in the way of facts. The other units at Chernobyl continued to operate for years and, as mentioned above, and in the link to my paper there were reasons that were preventable and controllable that influenced this disaster. Also, there was no nuclear explosion... it was a steam explosion that, because there was no containment building, allowed radioactive elements out and the moderator to catch fire, spewing even more radiation. The human element is compelling but a look at the facts shows that it should not have happened, in a western designed reactor would not have happened and if they had not cut cost by leaving out a containment building it wouldn't have affected anything anyway.
There are 400 something reactors operating in the world today. There are many that are decommisoned. Hundreds of research reactors span the globe. Of all these installations only one disaster that has compromised human health has occured. That's a staggering number and added to the environmental benefits described elsewhere on this site it tells a compelling picture: Nuclear power will save many more lives than it has caused from global warming and air pollution mitigation. It is esential as part of the energy mix as we head into a low emmision future.
Nuclear installations have an amazing safety record internally as well. Safety is number one priority around the world and the fact that these plants operate with very high capacity factors despite the fact they are shut down on the slightest indication of trouble is doubly amazing. The recordable incident rate is lower by around a factor of ten comparing the nuclear industry to manufacturing in general.