The Notch-Delta signalling pathway allows communication between neighbouring cells during development. It has a critical role in the formation of `fine-grained' patterns, generating distinct cell fates among groups of initially equivalent neighbouring cells and sharply delineating neighbouring regions in developing tissues. The Delta ligand has been shown to have two activities: it transactivates Notch in neighbouring cells and cis-inhibits Notch in its own cell. However, it remains unclear how Notch integrates these two activities and how the resulting system facilitates pattern formation. Here we report the development of a quantitative time-lapse microscopy platform for analysing Notch-Delta signalling dynamics in individual mammalian cells, with the aim of addressing these issues. By controlling both cis- and trans-Delta concentrations, and monitoring the dynamics of a Notch reporter, we measured the combined cis-trans input-output relationship in the Notch-Delta system. The data revealed a striking difference between the responses of Notch to trans- and cis-Delta: whereas the response to trans-Delta is graded, the response to cis-Delta is sharp and occurs at a fixed threshold, independent of trans-Delta. We developed a simple mathematical model that shows how these behaviours emerge from the mutual inactivation of Notch and Delta proteins in the same cell. This interaction generates an ultrasensitive switch between mutually exclusive sending (high Delta/low Notch) and receiving (high Notch/low Delta) signalling states. At the multicellular level, this switch can amplify small differences between neighbouring cells even without transcription-mediated feedback. This Notch-Delta signalling switch facilitates the formation of sharp boundaries and lateral-inhibition patterns in models of development, and provides insight into previously unexplained mutant behaviours.