A Neural Network is a mathematical object made of layers of simple functions.
Neurons and layers
Each unit takes a weighted sum of its inputs and passes the result through a non-linearity. Stack enough of those and you can approximate essentially any function.
Training
Training adjusts the weights so the model's outputs match a target. The technique that makes this efficient is called backpropagation.
The surprise of deep learning is not the depth. It is that scaling the same idea keeps working.