The "chain of command" is the rules for who's in charge of who in a company, a government department, the army, etc. In other words, it's the structure of relationships between superiors and subordinates, like bosses and their employees.
The phrase "chain of command" comes from the military, where there are strict rules about who is higher up (generals, etc) and lower (privates, sergeants) in the organization. The order is supposed to be like a chain from lowest to highest:
- Private > Sergeant > Lieutenant > Major > Colonel > General
To "follow" the chain of command means to follow the rules about who is allowed to make different decisions in an organization.