The steady expansion of neural recording capability provides exciting opportunities for discovering unexpected patterns and gaining new insights into neural computation. Realizing these gains requires flexible and accurate yet tractable statistical methods for extracting structure from large-scale neural recordings. Here we present a model for simultaneously recorded multi-neuron spike trains with negative binomial spiking and structured patterns of functional coupling between neurons. We use a generalized linear model (GLM) with negative-binomial observations to describe spike trains, which provides a flexible model for over-dispersed spike counts (i.e., responses with greater-than-Poisson variability), and introduce flexible priors over functional coupling kernels derived from sparse random network models. The coupling kernels capture dependencies between neurons by allowing spiking activity in each neuron to influence future spiking activity in its neighbors. However, these dependencies tend to be sparse, and to have additional structure that is not exploited by standard (e.g., group lasso) regularization methods. For example, neurons may belong to different classes, as is often found in the retina, or they may be characterized by a small number of features, such as a preferred stimulus selectivity. These latent variables lend interpretability to otherwise incomprehensible data. To incorporate these concepts, we decompose the coupling kernels with a weighted network, and leverage latent variable models like the Erd˝os-Renyi model, stochastic block model, and the latent feature model as priors over the interactions. To perform inference, we exploit recent innovations in negative binomial regression to perform efficient, fully-Bayesian sampling of the posterior distribution over parameters given the data. This provides access to the full posterior distribution over connectivity, and allows underlying network variables to be inferred alongside the low-dimensional latent variables of each neuron. We apply the model to neural data from primate retina and show that it recovers interpretable patterns of interaction between different cell types.