Automations allow you to gather resources and produce certain items even while you are not around, there are a few different automations and they come in two parts.
Part One
The room, automations can only go in specific rooms. For example a Lumber Mill automation can only go in a Lumber Mill room.
You can not put a Mine Shaft in a Farm for example.
Most of the Automations can be build as a standalone building (which comes with usually 4 rooms of the same type) or as a room deed which can be added on to an existing building.
Part Two
The furnishing, building the room is one thing, but to start producing materials you need a furnishing, these are typically named after the room name so there is a Lumber Mill furnishing that goes inside a Lumber Mill room, check out the Library > Items > Plans screen and view the Furnishings list for more information on the Furnishings available. As mentioned above these furnishings can only go inside the correct room. *Furnishing lvl dictates the lvl of the material produced.
Many of the furnishings require a certain level for use, this is because Automations are meant as a higher level item.
Use the standard building furnish screen to add your automation furnishing to the automation room, then click on the furnishing in the room and it will present you with a list of resources the automation can be set to make.
After choosing the automation is locked in and it will make only that resource, to change the resource you have to destroy the furnishing and place a new one.
A furnishing only has a limited capacity once the capacity has been met or exceeded the furnishing needs to be replaced, so furnishings do not offer unlimited resources you will have to replace them more often than not once every few days.
Also the higher your furnishing skill when you place an automation in a room the more capacity the furnishing has.
Visit your furnishing and click on it to empty out the resources it has created and remember to have more furnishing items in reserve to replace it with when it expires.
Automation produced resources are no different to normally produced resources. One automation will typically produce slower than a player but you can chain many automations together and given you can have 35+ buildings (at level 1000) the potential for resource manufacture per day is quite good.