A number of recent studies of information diffusion in social media, both empirical and theoretical, have been inspired by viral propagation models derived from epidemiology. These studies model the propagation of memes, i.e., pieces of information, between users in a social network similarly to the way diseases spread in human society. Importantly, one would expect a meme to spread in a social network amongst the people who are interested in the topic of that meme. Yet, the importance of topicality for information diffusion has been less explored in the literature. Here, we study empirical data about two different types of memes (hashtags and URLs) spreading through the Twitter's online social network. For every meme, we infer its topics and for every user, we infer her topical interests. To analyze the impact of such topics on the propagation of memes, we introduce a novel theoretical framework of information diffusion. Our analysis identifies two distinct mechanisms, namely topical and non-topical, of information diffusion. The non-topical information diffusion resembles disease spreading as in simple contagion. In contrast, the topical information diffusion happens between users who are topically aligned with the information and has characteristics of complex contagion. Non-topical memes spread broadly among all users and end up being relatively popular. Topical memes spread narrowly among users who have interests topically aligned with them and are diffused more readily after multiple exposures. Our results show that the topicality of memes and users' interests are essential for understanding and predicting information diffusion.