A top damper or cap damper is a metal spring door placed at the top of the chimney with a long metal chain that allows one to open and close the damper from the fire place.
One can purchase a chimney damper from a variety of stores. Chimney dampers can be purchased from fireplace specialists such as Woodland Direct, or from online stores such as Amazon.
Most dampers are designed to close DOWN, and are kept closed by the weight of the damper. Get a pair of safety glasses and a flashlight, check for foreign material or mortar that may be keeping the damper from seating, and that the metal of the damper plate is not warped.
fireplace
By flue, I believe you mean the damper in the chimney, which has the effect of closing off the flue. Close the damper any time you are not using the fireplace. Remember -heat rises -so heated air will also rise up and out the flue in the winter. In the summer it is also good to keep the flue closed because heat and humidity will activate the smelly properties of the ash and creosote in the firepalce chimney and create a bad odor in the home. In order for a damper to do its job it must seal tightly/ If your existing mechanical damper is broken, missing or in a bad state you can repair it or replace it with a chimney top damper or a chimney balloon. But, open the damper before lighting a a fire, or it will rapidly get quite smokey in the house.
FIREPLACE
When you use the fireplace, the damper should be open all the way. If not, you probably will not get enough updraft and your house will fill up with smoke...not a good thing. When it is not in use, the damper should be closed...otherwise you will lose a lot of heat up the chimney.
The flue is the actual pipe or chimney where the damper is the mechanism that opens and closes the flue
Your chimney damper is open
There are many kinds of dampers, depending upon how old the house is and the choice by the masons. Many dampers have a hinge on the rear and a handle that can be pushed or lifted to lift the front and latch it open. Others have a lever designed to move with a knob or chain connected to the damper. Some chimneys have counter-balanced dampers on the top of the chimney that are opened by pulling and latching a chain. A "tilt and slide" damper not only opens with a handle, but also has a movable grating that can be "slid" open and shut using a knob cast into it. If you have a gas log fireplace and can not close your damper by the handle it is very possible that the damper has been disabled or welded open according to the 1991 building code that states that dampers in new gas log fireplaces must be disabled. The only type of damper that is allowable to use in a gas log fireplace is an inflatable damper. If you do not have a gas log fireplace and you cannot open or close your damper it is likely that it has corroded to the point of non-function. The chimney is a very acidy and corrosive environment any metal in it will rust and degrade rapidly over time. Lubricate the damper with a penetrating oil and try to loosen it by hand or call a chimney sweep to come and work on it.
In the event of fire, the fusible link melts and the shutter gets closed. This helps in blocking the fire from spreading through the duct ----- If you are asking about a damper on a stove or chimney, that is not called a fire damper, but just a damper. There is a link to a question on that below.
If you are unable to open the damper in your fireplace, it may be due to a buildup of soot or debris obstructing its movement. You may need to clean the damper to ensure it opens and closes properly. If the damper still won't open, it could be a mechanical issue that requires professional inspection and repair.