Individual's decisions, from what product to buy to whether to engage in risky behavior, often depend on the choices, behaviors, or states of other people. People, however, rarely have global knowledge of the states of others, but must estimate them from the local observations of their social contacts. Network structure can significantly distort individual's local observations. Under some conditions, a state that is globally rare in a network may be dramatically over-represented in the local neighborhoods of many individuals. This effect, which we call the "majority illusion," leads individuals to systematically overestimate the prevalence of that state, which may accelerate the spread of social contagions. We develop a statistical model that quantifies this effect and validate it with measurements in synthetic and real-world networks. We show that the illusion is exacerbated in networks with a heterogeneous degree distribution and disassortative structure.