What is the difference between a neural network, artificial intelligence and an expert system?

A neural network is a computer system modeled after how we think the brain works. There are no emergent behaviors there, necessarily, but it is trained rather than programmed to solve problems. Artificial intelligence's holy grail is to produce a system that can pass a Turing test - the argument being that if an AI can convince someone that it is intelligent, then it IS intelligent. An expert system is just a piece of software programmed using artificial intelligence techniques. Such systems use databases of expert knowledge to offer advice or make decisions. An expert system will not pass a Turing test.