Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earthâ€™s surface. However, in modern contagions, long-range edges --- for example, due to airline transportation or communication media --- allow clusters of a contagion to appear in distant locations. We study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct "contagion maps" that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach also highlights contagion maps as a viable tool for inferring low-dimensional structure in networks.