How are the north and south celestial poles related to the earth's axis of rotation?
Astronomers need a coordinate system to define the positions of stars. Declination is the angle between the equator and the star, right ascension is the hour angle from the Vernal Equinox to the star.
The direction of the Vernal Equinox is the direction in the equatorial plane from the centre of the Earth to the Sun as it crosses the equator in March. The north celestial pole is then the direction from the centre of the Earth along the axis of rotation northwards.
The problem is that neither the Vernal Equinox direction or the direction of the Earth's axis of rotation are fixed. Both directions are constantly changing due to nutation caused by the gravitational effects of the Sun, Moon and other planets.
To get around this the coordinates of stars are given along with a reference date. A commonly used reference date is the J2000 epoch. The direction of the Vernal Equinox and the North celestial pole are then defined to be the directions as they were at 2000-01-01 12:00:00 TT (note the time is midday not midnight). Where TT is Terrestrial Time which differs from UTC by about a minute.
The current positions, the pole of date and the Vernal Equinox of date are calculated using power series which give the directions at any given time relative to the J2000 epoch.
So, the direction of the North celestial pole is the position that the Earth's axis of rotation was pointing at a particular moment in time such as the J2000 epoch 2000-01-01 12:00:00 TT.