Deep learning is a specific type of machine learning algorithm that uses multi-level neural networks.
A neural network is an algorithm that simulates the structure of the brain by having nodes (neurons) and relationships (dendrites). A neural network solves a problem by converting the problem into a set of input nodes, a set of output nodes, and a set of intermediate or "hidden" nodes. Deep learning is just a neural network with multiple intermediate layers of nodes.
To process input, the input values trigger the input nodes, which trigger the intermediate nodes, which attempt to solve the problem by triggering the correct output node.
Neural networks are normally implemented as a matrix of floating point numbers.
The first row in the matrix is the input values, the middle rows of the matrix are the intermediate layers, and the last row of the matrix is the output values.
For a given set of inputs, the intermediate node values are calculated by summing all of the previous node values using an "activation" function such as "segmoid". This is done for each layer until the output is produced. Since the matrix values are normally initialized with random numbers the output is random initially.
The network is then trained with validated data and the difference between the output and the correct output is used with "backpropagation" to correct the "weight" values in the intermediate nodes.
The goal is that with enough validated data, and enough layers, the network will eventually learn to produce values close to the correct output for most inputs.
For example to classify an image as a "dog" or "cat" the input could be the RGB pixels values of the image. The output would be the probability that the image is a dog or cat (2 outputs). With enough images, and a deep and complex enough network, it would be able to classify most images as a dog or cat.