Explainability is a topic of growing importance in NLP. In this work, we provide a unified perspective of explainability as a communication problem between an explainer and a layperson about a classifier's decision. We use this framework to compare several prior approaches for extracting explanations, including gradient methods, representation erasure, and attention mechanisms, in terms of their communication success. In addition, we reinterpret these methods at the light of classical feature selection, and we use this as inspiration to propose new embedded methods for explainability, through the use of selective, sparse attention. Experiments in text classification, natural language entailment, and machine translation, using different configurations of explainers and laypeople (including both machines and humans), reveal an advantage of attention-based explainers over gradient and erasure methods. Furthermore, human evaluation experiments show promising results with post-hoc explainers trained to optimize communication success and faithfulness.