Neural Viewer
NV simulates the behaviour of highly simplified biological neurons and renders them in realtime 3D using OpenGL. The software gives you the freedom to 'fly around' in space and inspect your network from all angles, and watch as individual spikes of electrical activation travel from one neuron to the next.

 
Free NN simulators
A large list of various NN simulators including downloads.

 
Perceptron Applet
This Java applet demonstrates a simple form of supervised learning entitled the perceptron learning algorithm. Using this applet, you can train the perceptron to act as a binary logic unit. It can compute or approximate most 2-input Boolean functions. However, a problem arises when trying to train the perceptron on the XOR (or XNOR) function. The applet provides a "work-around" for this problem by introducing an extra input.