A neural network is an interconnected group of biological neurons. In modern usage the term can also refer to artificial neural networks, which are constituted of artificial neurons. Thus the term 'Neural Network' specifies two distinct concepts:
An artificial neural network (ANN), also called a simulated neural network (SNN) or commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical or computational model for information processing based on a connectionist approach to computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network.
In more practical terms neural networks are non-linear statistical data modeling tools. They can be used to model complex relationships between inputs and outputs or to find patterns in data.
Please find the general introduction at Artificial neural network.
Neural network libraries -- Free Version
Neural network libraries -- Commercial version