I work with ANNs. However, your answer is too broad and imprecise for me to try to answer it completely in a few paragraphs. I agree with Walter that you should start with an elementary text and/or online sources like the comp.ai.neural-nets FAQ, wikipedia, etc.
The short answer is that typical ANNs are regression or classification algorithms that can be interpreted in terms of input,output and hidden nodes that are connected via multiplicative weights and activation functions.
The anachronym ANN indicates that the first significant work was w.r.t the (artifiical) modelling of biological neural networks using sigmoidal activation functions.
There are many types of ANNs and for each type there are many deterministic and random design approaches. Given a functioning ANN there is usually no way to determine how it was designed. However, sometimes you do get a hint (e.g., Genetic NN).
Hope this helps.
Greg
