The earth rotates about an axis which is inclined to the perpendicular to the ecliptic at an angle of 23 1/2°. Observations over centuries show that the axis of the earth itself keeps changing over the years. The rotation of the earth is like that of a spinning top, maintaining the same angle with the perpendicular to the ecliptic.
Have you ever observed the motion of a top?
When a top is set to spin and left to itself, it's axis of rotation keeps changing steadily, but somewhat slowly as compared to the spin. This continuous change in 'axis of rotation' is called as the precession of the top.
The earth is also like a spinning top. Its axis precesses around the perpendicular to the ecliptic. The axis of the earth is always perpendicular to its equator. So, as the axis keeps precessing around the perpendicular to the ecliptic, the equator also keeps changing its orientation. So does the celestial equator, which is in the same plane as the terrestial equator. But the ecliptic retains its same orientation in the space. Hence the point of contact of the ecliptic and the celestial equator, which are nothing but the equinoxes, changes. Precession of equinox is nothing but this change in the position of equinoxes along the ecliptic.
As a result of the change of the earth's axis, the 'Dhruva tara' which is the pole star now, will no longer be the pole star after some years, as the northpole will point to some other direction, and hence to some other star. After that some other star will appear stationary (above the changed pole) and will be called the Pole star. For instance, in 13,000A.D, 'Abhijit' will replace 'Dhruva tara' as the Pole Star.
It takes 26,000 years for the earth's axis to complete one rotation. Compare this with the period for completing one rotation around itself, i.e., one day. So the rate at which the earth's axis changes its direction is very small. It is very negligible over one year. Even though the change is small, it accumulates over centuries and one should take this into account for the precise description of celestial objects.
Say, at sometime the vernal equinox is at A and after one year the vernal equinox is at A1. This A1 will be to the west of A by a small shift of about 50". Similarly the autumnal equinox would also have shifted by 50" to the West. So if 'A' is the position of the vernal equinox now, it will be 90° from A after about 6500 years. After yet another 6500 years, the vernal equinox will shift further by 90°. So, the vernal equinox 13,000 years from now will coincide with the present autumnal equinox and after 26,000 years from now the vernal equinox will shift back to its present position at A after completing one cycle.