Adaline
Pattern Recognition: Classification of Digits 0-9


Backpropagation
Time-Series Forecasting: Prediction of the Annual Number of Sunspots


Hopfield
Autoassociative Memory: Associative Recall of Images


Bidirectional Associative Memory
Heteroassociative Memory: Association of Names and Phone Numbers


Boltzmann
Optimization: Traveling Salesman Problem


Counterpropagation
Vision: Determination of the Angle of Rotation


Self-organising map
Control: Pole Balancing Problem


Adaptive Resonance Theory
Brain Modeling: Stability-Plasticity Demonstration


 The ANNEvolve project
The ANNEvolve project is a hobby project of Mitchell Timin and Eric Anderson. One of their (open-source) programs is a NN which learns how to control a sailing boat. The source code is very well commented. A site worth checking out!