There are many many kinds of NNs by now. Nobody knows exactly how many. New ones (or at least variations of old ones) are invented every week. Below is a collection of some of the most well known methods, not claiming to be complete.
The two main kinds of learning algorithms are supervised and unsupervised.
In supervised learning, the correct results (target values, desired outputs) are known and are given to the NN during training so that the NN can adjust its weights to try match its outputs to the target values. After training, the NN is tested by giving it only input values, not target values, and seeing how close it comes to outputting the correct target values.
In unsupervised learning, the NN is not provided with the correct results during training. Unsupervised NNs usually perform some kind of data compression, such as dimensionality reduction or clustering.
