The earth rotates. It has an angular momentum in the direction from the south pole to the north pole. This axis of rotation, as I explained earlier, is tilted slightly, and this is what causes seasons. The axis of rotation, in fact, does not always tilt in the same direction. This direction moves slowly around in a circle every 25,800 years. This circular motion is called precession. First, I'll explain why it precesses, and then the implications of precession.
Why does the Earth precess?
First, see precession explained
There are two components of precession. First, there's some angular momentum, and second, there's some torque. The angular momentum of the earth is obvious: it spins, about once a day. The source of the torque is less obvious.
It comes from the Earth's equatorial bulge. The bulge is caused by the fact that at the equator, the surface of the earth moves much faster. When an object rotates quickly, what we call the centrifugal force pulls outward on the object. The earth doesn't rotate quickly enough to fling people out into space, but it does weaken gravity very slightly (by less than 1%). This causes the ground near the equator to be further from the center of the earth, making the equatorial bulge.
Because of the equatorial bulge, there is more mass on the earth's equator. The sun and moon both pull on this extra mass. They pull the equator towards the ecliptic plane. If the earth weren't rotating but still had a bulge, then it would be pulled until there was no axial tilt. However, since the earth is spinning, it causes precession instead. The axial tilt doesn't disappear, but instead moves around in circles, just like a wobbling gyroscope.
What are the effects of the Earth's precession?
Because the angular momentum changes direction, "north" changes direction. The north star is just some lucky star that happens to be directly north at the moment. However, over thousands of years, the direction north changes such that the north star is no longer north. There are several other stars that are "north stars" at different times in the precession cycle.
If you think of it from the star's point of view, the star used to point north, but then the world moved. But the world will come around again, eventually giving the star another moment. Perhaps this is symbolic of something?
There is another effect: a difference between the "sidereal year" and the "tropical year". A sidereal year is the amount of time it takes for the earth to complete a full orbit. A tropical year is the amount of time it takes to complete four seasons. The precession of the earth causes the axial tilt to rotate around in a circle every 25,800 years. The direction of the tilt relative to the direction to the sun causes seasons. That means that for every 25,800 sidereal years, there will be an extra four seasons. 25800 sidereal years equals about 25801 tropical years.
New Years and Astrology
Our calendar uses tropical years, not sidereal years, because it is more convenient to farmers, among other people. The end result is that when we celebrate the new year, we are not celebrating a complete orbit of the earth, back to the location at the beginning of the year. We're celebrating a rather arbitrary time marker. But that's ok. I can still wish you a Happy New Year!
I should also note that the positions of stars go by sidereal years, not tropical years. Since we've been using tropical years to count the last 2 millennia, the usual dates for astrological signs are actually off by about a month. Also, there's a new astrological sign in the northern hemisphere. Of course, most people never know this, probably because astrology is equally ineffective either way.