A circumpolar star is a star that, as viewed from a given latitude on Earth, never sets (it never disappears below the horizon), due to its proximity to one of the celestial poles. Circumpolar stars are visible for the entire night on every night of the year, and would be continuously visible throughout the day were they not overwhelmed by the Sun's glare.
As Earth spins daily on its axis, the stars appear to rotate in circular paths around one of the celestial poles (the north celestial pole for observers in the northern hemisphere, or the south celestial pole for observers in the southern hemisphere). Stars far from a celestial pole appear to rotate in large circles, stars located very close to a celestial pole rotate in small circles and hardly seem to engage in any diurnal motion at all. Depending on the observer's latitude on Earth, some stars, the circumpolar ones, are close enough to the celestial pole to remain continuously above the horizon, while other stars dip below the horizon for some portion of their daily circular path, and others remain permanently below the horizon.
The circumpolar stars appear to lie within a circle that is centered at the celestial pole and tangential to the horizon. At the Earth's North Pole, the north celestial pole is directly overhead, and all stars that are visible at all (that is, all stars in the northern celestial hemisphere) are circumpolar. As one travels south, the north celestial pole moves towards the northern horizon. More and more stars that are at a distance from it begin to disappear below the horizon for some portion of their daily orbit, and the circle containing the remaining circumpolar stars becomes increasingly small. At the Earth's equator this circle vanishes to a single point, the celestial pole itself, which lies on the horizon, and there are no circumpolar stars at all.
As one travels south of the equator the opposite happens. The south celestial pole appears increasingly high in the sky, and all the stars lying within an increasingly large circle centred on that pole become circumpolar about it. This continues until one reaches the Earth's South Pole where, once again, all visible stars are circumpolar.
The north celestial pole is located very close to the North Star (Polaris), so, from the northern hemisphere all circumpolar stars appear to rotate around Polaris. Polaris itself remains almost stationary, always at the north (the azimuth is 0°), and always at the same altitude (angle from the horizon), equal to the latitude of the point of observation on Earth.
The circumstances making a star circumpolar is solely dependant on the observer's hemisphere and their latitude. As the altitude of either the north celestial pole or south celestial pole is the same as the observer's latitude, any star whose position from the pole is less than the latitude, will be circumpolar and will never set below the horizon. If the observer latitude is 45°N and is facing north, then any star will become circumpolar if it lies less than 45° from the north celestial pole. If the observer's latitude is 35°S and is facing south, then these stars are circumpolar within 35° of the south celestial pole. Stars on the celestial equator will not be circumpolar when seen from any latitude in either hemisphere of the Earth.
This is easily calculate if some star will be circumpolar (or not) at the observer's latitude by just knowing the star's declination. Some stars within the far northern constellations, such as Cassiopeia, Cepheus, Ursa Major, and Ursa Minor, roughly north of the Tropic of Cancer (+23½°N), will be circumpolar stars that never rise or set.
Some Stars within the far southern constellations, such as Crux, Musca, and Hydrus, roughly south of the Tropic of Capricorn (-23½°S), will also be circumpolar stars.
Stars (and constellations) that are circumpolar in one hemisphere are always invisible in the high latitudes of the opposite hemisphere, and these never rise above the horizon. For example, the southern circumpolar star Acrux is invisible from most of the Continental United States, likewise, the seven stars of the northern circumpolar Big Dipper asterism are invisible from most of the Patagonia region of South America.
