backpropagation. An algorithm to compute the derivatives, needed for Gradient descent, for Artificial neural networks.

Backpropagaion. It effectively uses the chain rule to compute the gradient w.r.t. parameters at one layer with the values of the gradients w.r.t. parameters at the layer above (deeper).

Nice explanation

intro chapter

Backpropagation with shared weights

Efficient BackProp

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Video

See this too

Why backprop is more efficient than naive approach

Derivatives wrt the input give you a way of knowing which part of the input is determining the classification, i.e. where is the cat in the image, for example

Backprop in the brain (see Geoff Hinton vid) and this paper: http://biorxiv.org/content/early/2016/12/23/035451

STDP-Compatible Approximation of Backpropagation in an Energy-Based Model