Most commonly, NN's use supervised training.
Many algorithms exist that perform supervised training.
In each case every input vector is associated with a target output vector. The actual output of the net may or may not match the target output, depending on the weights at that particular moment. The earliest and simplest learning rule for a neural net is generally known as the Hebb rule. Inputs and outputs may be in binary or bipolar form, but binary does not work as well since the net never learns if the target is 0. The Hebb rule is limited in that it is not guaranteed to give correct outputs after training.