Classic network clustering tackles the problem of identifying sets of nodes (communities) that have similar connection patterns. However, in many scenarios nodes also have attributes that are correlated and can also be used to identify node clusters. Thus, network information (edges) and node information (attributes) can be jointly leveraged to design high-performance clustering algorithms. Under a general model for the network and node attributes, this work establishes an information-theoretic criteria for the exact recovery of community labels and characterizes a phase transition determined by the Chernoff-Hellinger divergence of the model. The criteria shows how network and attribute information can be exchanged in order to have exact recovery (e.g., more reliable network information requires less reliable attribute information). This work also presents an iterative clustering algorithm that maximizes the joint likelihood, assuming that the probability distribution of network interactions and node attributes belong to exponential families. This covers a broad range of possible interactions (e.g., edges with weights) and attributes (e.g., non-Gaussian models) while also exploring the connection between exponential families and Bregman divergences. Extensive numerical experiments using synthetic and real data indicate that the proposed algorithm outperforms algorithms that leverage only network or only attribute information as well as recently proposed algorithms that perform clustering using both sources of information. The contributions of this work provide insights into the fundamental limits and practical techniques for inferring community labels on node-attributed networks.