Your question is not specific, so I can only give a general answer.
Most, if not all, microcontrollers that are used in these SBC (Single Board Computers) can go into low-power mode. They switch to a much lower clock speed and that causes the reduction in power.
They either wake up periodically for a very short period of time to check their inputs or external events can cause them to go into full-power mode. External events can be a button press or receiving something on a communication channel.
We, for instance, make modules that are use in power-wheelchairs (using ARM microcontrollers) that can wake up due to activity on the CAN bus. This way, one module can wake up the whole network.
If you want to achieve something similar, then you will have to look for low-power mode support in the development environment you are using.