Paraprosdokian is a word which comes to us from Greek; it can be broken down into two parts:

para: beside or against
prosdokia: expectation

In the case of our word, we should use the meaning "against" for the prefix para. Thus, the word paraprosdokian means "against expectation." A paraprosdokian is a phrase or statement in which the conclusion of the statement causes you to rethink what was said in the first part of the statement -- often for a surprising or humorous effect. 

A famous example of a paraprosdokian is Groucho Marx's statement which begins: "I've had a perfectly wonderful evening..." and then ends, "...but this wasn't it." The humor lies in the fact that, as Groucho begins his statement, you think he is saying that this was a wonderful evening, but with the addition of four more words, the meaning is completely changed, and we're left with the realization that he is implying that this was a perfectly horrendous evening.

Many comedians besides Groucho Marx made use of the paraprosdokian; for example, the following is attributed to Stephen Wright: "I want to die peacefully in my sleep, like my grandfather; not screaming and yelling like the passengers in his car." The first part of the sentence conjures up the peaceful image of an elderly man lying peacefully on a bed with his eyes closed, and a clean white sheet tucked under his chin. Then the second half of the sentence comes, and the image abruptly changes to one of horror and mayhem. The surprising nature of the change in imagery makes us laugh, despite the imagery of carnage and wreckage.

When we took our first child to the pediatrician for one of his early well-child checkups, the doctor looked at him, then looked at me, and then said, "I can see he got his good looks from you..." In the brief pause she left before finishing her sentence, I was horrified that she would say such a thing with my wife standing right there. Then she finished: "...because your wife clearly still has hers." That was a paraprosdokian.

Incidentally, many haiku are  paraprosdokians. One of the defining characteristics of haiku, which is often left out when being taught in western schools, is that the haiku is intended to present one image, and then cause the reader to rethink that image in the conclusion of the poem.