Artificial neural networks (ANNs) are a broad class of computational models loosely based on biological neural networks. They encompass feedforward NNs (including "deep" NNs), convolutional NNs, recurrent NNs, etc.
Artificial neural networks (ANN), are composed of neurons --- programming constructs that mimic the properties of biological neurons. A set of weighted connections between the neurons allows information to propagate through the network to adapt to specific tasks automatically, without the designer hand-crafting inputs and features.
Neural networks have a wide variety of uses and flavors. Some prominent artificial neural network types are perceptron, conv-neural-network, rnn (e.g. lstm). Modern feedforward neural networks are often comprised of many "layers", and often termed "deep networks" (a.k.a. "deep learning"), in contrast to so-called "shallow" networks comprised of a single hidden layer. Sometimes neural networks are used to train computers to learn strategies, e.g. in chess, Go, or video games; these models use reinforcement-learning approaches, e.g. q-learning.
Among freely available, high-quality resources on neural networks, the Deep Learning book by Ian Goodfellow, Yoshua Bengio, and Aaron Courville can be mentioned.