This is a very slow movement of Earth’s polar axis. Over a human lifetime it always appears to point in the same direction in space – towards the Pole Star – so the stars all move around the Pole Star during their daily movement. But in fact the axis is moving very slowly all the time, and over the centuries the starry background changes position. If you could see the change happening over thousands of years, you’d see that if you were looking at midnight of 1 January, for example, you’d see that although the Sun, Moon and planets always move along the same line in the sky, known as the ecliptic, the starry background slips along the same line. This has the effect that the constellations visible at a particular time of year change, and some stars that are currently always below the horizon as seen from the UK, such as the Southern Cross, become visible. The whole cycle takes 26,500 years before the sky is back again where you started.
Precession alters the positions of the stars year by year, because they are measured from a point on the ecliptic that relates to Earth’s orbit around the Sun, and when you quote the position of the star you need to mention what year it refers to (known as the epoch). Currently, positions are measured to epoch 2000.0, though for small and fast-moving objects such as comets and asteroids the epoch of date is more useful so that you can see the position of the object against the current starry background.