The Fast Artificial Neural Network Library is a programming library that takes much of the pain out of constructing artificial intelligence and cognitive modelling projects.
It is free software, incredibly professional, well documented, fully supported, and available for a number of programming languages both mainstream and obscure.
There’s also a concise introduction to neural networks (pdf) which covers some of the operating principles for those wanting to know how they work.
Neural networks are used both as software tools for completing otherwise difficult tasks, and in cognitive science for simulating cognitive processes.
In neuropsychology, neural networks are often created to simulate a certain cognitive task, and then the network is ‘damaged’ to see whether the network can predict the effects of brain injury or impairment.
This connectionist approach to cognitive science was made particularly popular by the 1986 book Parallel distributed processing: Explorations in the microstructure of cognition (ISBN 0262631121) by David Rumelhart and James McClelland.