[oldspiceman]

A less mathy explanation with some real examples: http://neuralnetworksanddeeplearning.com/chap1.html

Coding a digit recognizer using a neural network is an extremely rewarding exercise and there's a lot of help on the web to get you started.

[agibsonccc]

This is a great example of a hello world application. Keep in mind there are several kinds of neural nets that allow you to do this. This includes convolutional RBMs (recognizes parts of an image) and normal RBMs (learns everything at once)