Here is a quick and dirty summary. I was not in ops so it should be high level enough for normal people to understand, but dumb enough that an ops guy would jump my case.
The time to cold shutdown is dependent on many things, but mainly decay heat. Basically, it's when the coolant is maintained below boiling at atmospheric pressure. If a unit is near the end of its cycle, cold shutdown may be reached relatively quickly, but if a unit has to be taken down early in the cycle for some reason, it will take much longer. If all of the safety systems aren't working, it may take longer to cool. At Fukushima, the reactor cooling systems became inoperable, which allowed the coolant to boil off, causing core meltdown.
PWRs generally have more parts that operate at higher pressures, but the secondary side (steam) is not irradiated. This reduces dose and makes things on the secondary side easier to work on. They do require steam generators which are extremely expensive...an error in design of replacement SGs permanently shutdown San Onofre due to the prohibitive costs to fix the error. PWRs also use boron to control reactivity, a system that can be a pain in the ass to maintain. Ps usually have more generating capacity.
BWRs usually have fewer parts, but the steam is produced in the reactor and is irradiated. This results in more dose and things are a pain to replace/maintain/etc due to contamination (crapped up). Bs do not use boron except as a last resort...as in permanent shutdown. Bs usually have less generating capacity.
The general concept for each type is the same. Keep the core covered by keeping your safety systems operable. US plants do this by "defense in depth". Basically, each safety system has more than one "train" so if one goes down, you have another one or more as backup. Some systems are redundant. The plant is designed in such a way that in a design basis event (accident) the systems initiate in a controlled manner at certain set points to maintain core integrity. The Senior Reactor Operators and their crews know every detail of the units and how they operate. The must go through rigorous training to become licensed and ongoing training every 6, 9, ? (don't remember) weeks to maintain their license.
That's based on my limited experience at the fleet level of a utility having both Ps and Bs. But I've escaped that world.