Recent experimental advances are producing an avalanche of data on both neural connectivity and neural activity. To take full advantage of these two emerging datasets we need a framework that links them, revealing how collective neural activity arises from the structure of neural connectivity and intrinsic neural dynamics. This problem of structure-driven activity has drawn major interest in computational neuroscience. Existing methods for relating activity and architecture in spiking networks rely on linearizing activity around a central operating point and thus fail to capture the nonlinear responses of individual neurons that are the hallmark of neural information processing. Here, we overcome this limitation and present a new relationship between connectivity and activity in networks of nonlinear spiking neurons. We explicitly show how recurrent network structure produces pairwise and higher-order correlated activity, and how nonlinearities impact the networks' spiking activity. Finally, we demonstrate how correlations due to recurrent connectivity impact the fidelity with which populations encode simple signals and how nonlinear dynamics impose a new effect of correlations on coding: to translate response distributions in addition to stretching them. Our findings open new avenues to investigating how neural nonlinearities---including those expressed across multiple cell types---combine with connectivity to shape population activity and function.