There is nothing more important than your Headquarters. Currently it is the only building that provides you with Command Points so it is required to build units. But that is not the only thing it can do. Besides providing energy for your buildings it also produces a small amount of resources.
Normally you only need the one provided to you at the start of the game even though you need to upgrade it to get access to a higher Command Point Limit. Higher HQ Levels also give access to better Units and Defenses.
Each Headquarter you build produces the exact same amount of resources and Energy as the previous one on each level. This makes them an alternative to additional Supply Centers with the added bonus of producing instead of using energy and being a backup Headquarters if your first one gets destroyed. Only build them if any of the following requirements are met:
You are at least in Gold League (Rank 16+)
You already have at least 8 supply centers built
You have too many resources already
Your Superweapon is not a viable option for some reason