general relativity - Penrose Diagrams Null Coordinates

I am an engineering student interested in astrophysics. I started to watch a video about Penrose diagrams, since I wanted to find out how they work. Around the seventh minute the lecturer in the video said that in order to create a Penrose Diagram "non-compact"coordinates (where at least one goes against infinity) are replaced by null coordinates (which are still non-compact). He then went ahead and defined a coordinate function $u$ as null coordinate if its $$g(\frac{\partial}{\partial u},\frac{\partial}{\partial u})=0.$$ He also noted that such coordinates are "light-like". Well, I have a hard time to understand what he means. First, I am not familiar with his notation: I assume $g$ is the divergence? But the divergence of what field? Why does light have the property of divergence equals zero? I would be grateful for some clarification.