An enthymeme is a truncated argument in which either one of the premises, or the conclusion, is left unstated. Some examples:
Since McAuliffe is a Democrat, he is mendacious. This is an argument whose validity requires the tacit premise, ‘All Democrats are mendacious’ a proposition that is clearly false. This argument is an enthymeme since it is left to the consumer of the argument to supply the tacit (major) premise.
Since all Democrats are mendacious, McAuliffe is mendacious. This is an argument whose validity requires the tacit (minor) premise, ‘McAuliffe is a Democrat.’
Democrats are mendacious, and McAuliff is a Democrat. Here the consumer is invited to supply the conclusion, ‘McAuliffe is mendacious.’
Capital punishment is wrong because it involves killing a human being. When we supply the tacit premise, without which the argument is invalid, we see that the argument is unsound, precisely because of the falsity of the premise that must be supplied: ‘All killing of human beings is wrong.’ The counterexamples to this are obvious: killing in self-defense, and killing in just wars.
(A sound argument is a deductive argument whose form is valid and whose premises are true. An unsound argument is therefore either (i) one whose form is invalid, or (ii) one that has has one or more false premises, or both (i) and (ii)).
Careful thinkers and writers either avoid enthymemes or else use them sparingly: they argue explicitly, laying all of their assumptions on the table, so that both they and their readers can see if the arguments are cogent. In the capital punishment example, the lack of cogency springs to the eye once the argument is set forth explicitly.