A new era of Information Warfare has arrived. Various actors, including state-sponsored ones, are weaponizing information on Online Social Networks to run false information campaigns with targeted manipulation of public opinion on specific topics. These false information campaigns can have dire consequences to the public: mutating their opinions and actions, especially with respect to critical world events like major elections. Evidently, the problem of false information on the Web is a crucial one, and needs increased public awareness, as well as immediate attention from law enforcement agencies, public institutions, and in particular, the research community. In this paper, we make a step in this direction by providing a typology of the Web's false information ecosystem, comprising various types of false information, actors, and their motives. We report a comprehensive overview of existing research on the false information ecosystem by identifying several lines of work: 1) how the public perceives false information; 2) understanding the propagation of false information; 3) detecting and containing false information on the Web; and 4) false information on the political stage. In this work, we pay particular attention to political false information as: 1) it can have dire consequences to the community (e.g., when election results are mutated) and 2) previous work show that this type of false information propagates faster and further when compared to other types of false information. Finally, for each of these lines of work, we report several future research directions that can help us better understand and mitigate the emerging problem of false information dissemination on the Web.