Back-propagation, an introduction

Given the sheer number of backpropagation tutorials on the internet, is there really need for another? One of us (Sanjeev) recently taught backpropagation in undergrad AI and couldn't find any account he was happy with. So here's our exposition, together with some history and context, as well as a few advanced notions at the end. This article assumes the reader knows the definitions of gradients and neural networks. It is the basic algorithm in training neural nets, apparently independently rediscovered several times in the 1970-80's (e.g., see Werbos' Ph.D. thesis and book, and Rumelhart et al.).