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Driven by the outstanding performance of neural networks in the structured Euclidean domain, recent years have seen a surge of interest in developing neural networks for graphs and data supported on graphs. The graph is leveraged at each layer of the neural network as a parameterization to capture detail at the node level with a reduced number of parameters and computational complexity. Following this rationale, this paper puts forth a general framework that unifies state-of-the-art graph neural networks (GNNs) through the concept of EdgeNet. An EdgeNet is a GNN architecture that allows different nodes to use different parameters to weigh the information of different neighbors. By extrapolating this strategy to more iterations between neighboring nodes, the EdgeNet learns edge- and neighbor-dependent weights to capture local detail. This is a general linear and local operation that a node can perform and encompasses under one formulation all existing graph convolutional neural networks (GCNNs) as well as graph attention networks (GATs). In writing different GNN architectures with a common language, EdgeNets highlight specific architecture advantages and limitations, while providing guidelines to improve their capacity without compromising their local implementation. An interesting conclusion is the unification of GCNNs and GATs -- approaches that have been so far perceived as separate. In particular, we show that GATs are GCNNs on a graph that is learned from the features. This particularization opens the doors to develop alternative attention mechanisms for improving discriminatory power.