There's an out of control trolley speeding toward a group of people. If it rushes in amongst them, it will kill a number of them and injure others. You are near a switch that would allow you to redirect the trolley away from those people, onto a track where there's only one person. Do you pull the switch?What we learn from the problem is that some people feel very strongly that it would be wrong to pull the switch, because that implicates them in guilt for killing the one man. The world as they find it is not their fault, but electing to act means taking responsibility for the choice. Thus, they will let many people die to avoid being personally guilty for one death.
Others -- myself included -- feel that not acting is also a choice, and the desire to avoid responsibility is thus a false choice. Even here, moral intuitions differ. Some will pull the switch, believing it better to choose to save more lives. Others will refuse, believing that their chief duty is to refuse to commit murder. Roughly speaking, these choices break you out into the two leading contemporary schools of ethics, consequentialism (i.e., that morality means doing what has the best consequences for the most people) and deontology (i.e., that morality means doing your duty).
Now that I've told you all that, in case any readers weren't familiar with it, we can all enjoy the joke together.
UPDATE: Still more variations.