Artificial Neural Network
Artificial Neural Networks are in silco versions of neural networks. They are notable as compuational models for being inspired by how the brain works and presents a computational model which operates on different principles from those most (software and hardware) engineers are familiar with. (Specifically, artificial neural networks are not programmed.)
Learn By Example
An important characteristic of neural networks, including artificial neural networks, is that instead of programming to solve problems (like with Von Neumann machines) neural networks "learn by example" to solve problems.
In situations where one has little or an incomplete understanding of the problem to solve, the "learn by example" computational model which neural networks provide make them very appealing.
Artificial neural networks are made up of artificial neurons which are inspired by biological neurons. There are many different types of artificial neurons. Some artificial neurons aim for bioligical realism, while others simply strive for alternate computational models which happen to be loosly inspired by biological neurons.
Parallel Computational Model
Artificial neural networks are also intrinsically a parallel computational model, which can allow for fast computation of solutions which implemented on a parallel hardware architecture in silco.