A circumpolar star is a star that, as viewed from a given latitude on Earth, never sets (that is, never disappears below the horizon), due to its proximity to one of the celestial poles. Circumpolar stars are therefore visible (from said location) for the entire night on every night of the year (and would be continuously visible throughout the day too, were they not overwhelmed by the Sun's glare).
As the 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 hence 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 centred 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 therefore effectively 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 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 (i.e., 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.

Alternative definition
Some astronomers define a circumpolar star as any star having a declination of 55 or more degrees from the celestial equator; i.e., a declination between 55° and 90° on the celestial sphere. In other words, circumpolar stars are within 35° of the north or south celestial pole. These astronomers propose this idea because, from the vantange point of stargazers outside of the tropics, most stars with declinations between 55° and 90° never rise or set on the horizon in their respective hemispheres.
Stars of extreme northern constellations, such as Cassiopeia, Cepheus, Ursa Major, and Ursa Minor are well within this zone. Observers living in mid-northern latitudes always see the stars of these constellations above the horizon, never rising or setting.
Stars of extreme southern constellations, such as Crux, Carina, and Hydrus are also well within this zone of circumpolar stars. Observers living in mid-southern latitudes, in countries such as Australia, South Africa, and Argentina, always see the stars of these constellations above the horizon. From their vantange point, the diurnal circles of these extreme southern stars do not cross below the horizon; therefore, they never rise or set.
Stars and constellations that are circumpolar (within 35° of the celestial pole) in one hemisphere are virtually invisible, never rising above the horizon, in the middle and high latitudes of the opposite hemisphere. For example, the southern circumpolar star Acrux is invisible from most 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.