Precession is due to the fact that the resultant of the angular velocity of rotation and the angular veolcity produced by the torque is an angular velocity about a line which makes an angle with the permanent rotation axis, and this angle lies in a plane at right angles to the plane of the couple producing the torque. The permanent axis must turn towards this line, since the body cannot continue to rotate about any line which is not a principle axis of maximum moment of inertia; that is, the permanent axis turns in a direction at right angles to that in which the torque might be expected to turn it. If the rotating body is symmetrical and its motion unconstrained, and if the torque on the spin axis is at right angles to that axis, the axis of precession will be pependicular to both the spin axis and torque axis. Under these circumstances the period of precession is given by:
In which Is is the moment of inertia, Ts is the period of spin about the spin axis, and Q is the torque. In general the problem is more complicated than this, however.
The Earth's axis undergoes precession due to a combination of the Earth's nonspherical shape (it is an oblate spheroid, bulging outward at the equator) and the gravitational tidal forces of the Moon and Sun applying torque as they attempt to pull the equatorial bulge into the plane of the ecliptic. It goes through one complete precession in a period of approximately 25,800 years during which the positions of stars within the celestial sphere will slowly change. Over this period, the axis' north pole moves from where it is now, within 1° of Polaris, in a circle. Polaris isn't particularly well-suited for marking the north celestial pole, as its visual magnitude is only 1.97, fairly far down the list of brightest stars in the sky. On the other hand, in 3000 BC the faint star Thuban in the constellation Draco was the pole star; at magnitude 3.67 it is five times fainter than Polaris, and all but invisible from light-polluted urban skies. The brightest star to be North Star at any time in the forseeable past or future is the brilliant Vega, which will be the pole star in 14000 AD.
Polaris is not exactly at the pole; any long exposure unguided shot will show it having a short trail. It's close enough, though. The south celestial pole precesses too, always remaining exactly opposite the north pole. The south pole is in a particularly bland portion of the sky, and the nominal south pole star is Sigma Octantis, which, while fairly close to the pole, is even weaker than Thuban -- magnitude 5.5, which is barely visible even under a properly dark sky. The precession of the Earth is not entirely regular due to the fact that the Sun and Moon are not in the same plane and move relative to each other, causing the torque they apply to Earth to vary. This varying torque produces a slight irregular motion in the poles called nutation.
Precession of the Earth's axis is a very slow effect, but at the level of accuracy at which astronomers work, it does need to be taken into account. Note that precession has no effect on the inclination ("tilt") of the plane of the Earth's equator (and thus its axis of rotation) on its orbital plane. It is 23.5 degrees and precession does not change that. The inclination of the equator on the ecliptic does change due to gravitational torque, but its period is different (main period about 41000 years).
Hipparchos first estimated Earth's precession around 130 BC using his own observations and those of other greek astronomers in the preceding centuries. In particular they measured the distance of the stars like Spica to the Moon and Sun at the time of lunar eclipses, and because he could compute the distance of the Moon and Sun from the equinox at these moments, he noticed that Spica and other stars had moved over the centuries.
Precession causes the cycle of seasons (tropical year) to be a few minutes less than the cycle of the sun as seen with respect to the stars (sidereal year). This results in a slow change in the position of the sun with respect to the stars at an equinox. It is significant for calendars and their leap year rules.
The orbit of a planet around the Sun is also a form of rotation, and so the axis of a planet's orbital plane will also precess over time. Discrepancies in the precession rate of the planet Mercury compared to those predicted by classical mechanics were one of the major pieces of evidence leading to the acceptance of Albert Einstein's Theory of Relativity, which predicted the anomalies accurately.