An Idiot's Guide to Neural Networks

If you use the 'Search' option on your browser to look for articles on Neural Networks or "Connectionism" (which is another name for the same subject), you will find a great many sites explaining what they are and how they work. Unfortunately, they all seem to be written by mathematicians, all of whom speak Double Dutch. This is fine, providing you speak fluent higher mathematics, but when I wanted to find out how they worked, I looked from a programmer's point of view. What if I really wanted to find program source code, would I find any? Here, the idiots guide to neural networks comes to aid.