Human languages share many grammatical properties. We show that some of these properties can be explained by the need for languages to offer efficient communication between humans given our cognitive constraints. Grammars of languages seem to find a balance between two communicative pressures: to be simple enough to allow the speaker to easily produce sentences, but complex enough to be unambiguous to the hearer, and this balance explains well-known word-order generalizations across our sample of 51 varied languages. Our results offer quantitative and computational evidence that language structure is dynamically shaped by communicative and cognitive pressures.