Block diagrams give us graph-like ways to visually represent systems, especially control systems. They’re important because they allow us to model sub-systems and their dependencies, and how inputs flow to outputs.
Translating between complicated block diagrams and closed form transfer functions is possible, but somewhat time costly.
Basic configurations
A cascaded connection gives us: A parallel connection gives us: A feedback connection gives us:
Block manipulation
There are also several other ways we can manipulate blocks to give us an easier block diagram to simplify into a single transfer function.