There are many algorithms behind Deep Learning (see this comparison of deep learning frameworks for details), but one common algorithm used by many frameworks is Convolutional Neural Networks (CNNs). The mathematics behind that algorithm are complex, but Brandon Rohrer explains the process in plain language, and shows how AIs trained with CNNs can appear to mimic human processes like vision:
You can also download the slides from Brandon's presentation for offline viewing. Check out in particular Slide 90 which includes links to several software frameworks for deep learning, including Microsoft's open-source CNTK toolkit.
For a text-based explanation of CNNs from Brandon, follow the link to his post on KDnuggets below.
KDnuggets: How Convolutional Neural Networks Work