A fixed star is a celestial object that does not seem to move (in comparison to the other stars of the night sky). Hence, a fixed star is any star except for the Sun. Such 'fixed' stars do have parallax, but this was small enough to not have been noticed until modern times.
This name dates back from the time when it was not yet discovered that there were two differences between the planets and the stars. Planets seem to move, while stars do not, but also the planets (and their moons) only reflect the Sun's light, while the stars provide their own light.
