Neural networks are artificial systems inspired by biological neural networks that learn from data without being explicitly programmed. They work by analyzing datasets to identify characteristics and make predictions. For music recommendations, AI algorithms learn a user's music preferences from streaming history and other data, and then analyze songs to find similar music to suggest new songs and artists the user might like. Neural networks use layers like dense layers that perform operations to change vector dimensions and apply transformations. Weights, biases, and activation functions are important concepts that determine how neural networks process and output information.