Humor is an essential human trait. Efforts to understand humor have called out links between humor and the foundations of cognition, as well as the importance of humor in social engagement. As such, it is a promising and important subject of study, with relevance for artificial intelligence and human-computer interaction. Previous computational work on humor has mostly operated at a coarse level of granularity, e.g., predicting whether an entire sentence, paragraph, document, etc., is humorous. As a step toward deep understanding of humor, we seek fine-grained models of attributes that make a given text humorous. Starting from the observation that satirical news headlines tend to resemble serious news headlines, we build and analyze a corpus of satirical headlines paired with nearly identical but serious headlines. The corpus is constructed via Unfun.me, an online game that incentivizes players to make minimal edits to satirical headlines with the goal of making other players believe the results are serious headlines. The edit operations used to successfully remove humor pinpoint the words and concepts that play a key role in making the original, satirical headline funny. Our analysis reveals that the humor tends to reside toward the end of headlines, and primarily in noun phrases, and that most satirical headlines follow a certain logical pattern, which we term false analogy. Overall, this paper deepens our understanding of the syntactic and semantic structure of satirical news headlines and provides insights for building humor-producing systems.