The Controller is the core of the Refined Storage network. You need to connect all the Refined Storage devices to this block.
You are supposed to connect every device that depends on the storage network connection to the Controller.
You can connect devices with cables.
Cables are not always necessary, as long as the devices are next to each other in any direction a connection will be established as well.
The Controller accepts and consumes Forge Energy (FE). Most mods have a way to generate energy, therefore Refined Storage doesn't come by itself with a way to generate energy.
The energy is extracted per tick. The amount of energy that is extracted depends on the amount of devices, and which devices. Every device type consumes another amount of energy. You can see how many energy each device consumes by right clicking the Controller.
If the Controller runs out of energy, the network will be turned off until there is sufficient energy provided again.
When the Controller is broken, it will maintain its energy.
Each Refined Storage network can only have a single Controller. It doesn't make sense to connect multiple controllers on the same network.
If you connect another Controller to an already existing network, the Controller will break.