A damper is sort of like a valve in the flue that controls the flow of exhaust gasses. These days the term is a bit ambiguous because there are really three different sorts of dampers, that serve three different functions.
The older traditional type may or may not be needed with a modern stove, but was very important on older stoves. It is typically either at the exit from the stove or somewhere in the flue passage. It controlled the effective size of the flue, and thus restricted the total airflow through the stove. On a modern stove, it is usually not needed unless the flue has an unusually powerful draft that can override the modern stoves normal controls. In that case, adding a damper can restrict the effective flue volume so that the stove works better. (Draft, or draw, is the force that pulls the smoke up the chimney, it is influenced by many things, including the height, construction and size of the flue, the outside temperature, how hot the stove is, etc. )
In a modern stove, there are several techniques used to make it more efficient and cleaner burning. Many of these involve restrictions and extra chambers for secondary combustion, which work well once the stove is burning well, but can be too restrictive when the stove is being lit, or when you open it up to add more wood, so that the stove would blow smoke into the room. To deal with this, many stoves use a bypass damper that is usually internal to the stove body, and which (as the name implies) opens and closes a larger passage around those restrictions so that the stove wont smoke when you open the doors, and allows you to get it lit or get reloads burning well. This is one of the main controls on the stove itself.
In the older "Airtight" stoves, there were often additional dampers located on the stove doors or elsewhere on the stove body that regulated the flow of combustion air going INTO the stove. These acted in combination with the exhaust dampers to regulate the level of combustion inside the stove. In a more modern stove, there are normally two air intake sources, which allow somewhat less control over the combustion air, but ensure cleaner burning. The PRIMARY air supply is typically adjustable by the user, and can control the level of combustion to some degree. It can be very important not to leave the primary air supply open too far as this can cause the stove to over fire, however typically one should have the primary air open when lighting or reloading the stove, gradually closing it as the new load of fuel warms up and gets fully ignited. The exact details of this will vary with the stove. In addition, there is also a SECONDARY air supply, that can be either a separate air inlet or a "stop" on the primary that doesn''t allow it to be closed all the way. The secondary air supply may either be fixed or automatically regulated by the stove (depending on model) and is intended to promote clean burning by guaranteeing the fire a certain minimum supply of air.