From the user perspective, they are implemented as various texture maps to define base color, roughness, metalness and specular. Base color map is as it sounds. Roughness map is a map that goes from full reflective to dull. In practice it can be calculated similar to and stored the same as Mip Map. Metallic map can go from mirror like to non mirror in degrees. The lesser used Specular map lets you control the specular light property in a non metallic surface. From non reflective to full reflective. Specular is called Cavity in the linked document and also explained as typically an Ambient Occlusion map with a very short distance. "Used for small-scale shadowing."

From the real-time light equation that runs in the GPU shaders, all the details can be found in the 2013 Siggraph talk by Brian Karis of Epic Games. https://cdn2.unrealengine.com/Resources/files/2013SiggraphPr...

The shader code is on Github, however you have to sign up first to get access.