We present a framework with minimal ingredients for the study of coevolution in dynamical networks. This phenomenon is observed in many complex systems in nature and consists of the coexistence of two processes on networks of interacting elements: node state change and rewiring of connections between nodes. We consider that the process by which a node changes its connections, called rewiring, and that the process by which a node changes its state, can have their own dynamics, characterized by probabilities Pr and Pc that express the time scales for each process, respectively. We describe the process of rewiring in terms of two basic actions: disconnection and reconnection between nodes, both based on a mechanism of comparison of their states that in turn occur with different probabilities, d and r. Then, for a given rewiring process, a coevolution model corresponds to a specific coupling relation Pc = f(Pr). The collective behavior of a coevolutionary system can be studied on any subspace of the parameters Pr, Pd, d, and r. As an application, for a voterlike node dynamics we find that reconnections between nodes with similar states lead to network fragmentation. The critical boundaries for the onset of fragmentation in networks with different properties are calculated on the space (Pr,Pc). The occurrence of a homogeneous connected phase and a fragmented phase in a network is predicted for diverse models, and agreement is found with some earlier results. We also find regions of parameters where modular structures with nodes in different states coexist for very long times on one large, connected network. Thus, coevolution under some conditions can be seen as a mechanism for the emergence of communities observed in many real networks.