A coelostat is a device consisting of two mirrors that follow the movement of astronomical objects and reflect their image into a telescope. 'Coelostat' may refer to the set of flat mirrors (one fixed and one movable) and whatever apparatus is used to manipulate them, or the mirror-telescope combo. The movable mirror follows the desired object across the sky, reflecting it back into the fixed mirror. The second mirror, in turn, reflects the image into the telescope. (A second mirror is used so that the image is not backwards when reflected into the telescope). A focusing mirror may also be used.
The basic advantage of using a coelostat is that when looking at objects that change position over time, you are only changing the position of one mirror, instead of the entire telescope. This becomes useful in observatories that have very large telescopes.
These days coelstats are most often used for observing the sun, as it has a very predictable movement which may be of interest of a long period. In old science fiction novels, coelstats pop up all the time, as every object you might want to view will be moving in relation to a moving spaceship. Nowadays we assume the computer will take care of it.
Coelostat comes from the Latin word meaning 'sky-stopper'.
Other mirror configurations that you might use in an astronomical observatory are the heliostat and the siderostat.