The celestial equator is part of a larger system called the celestial sphere that is used as a coordinate system for locating and describing the position of celestial objects. The celestial sphere is not a physical object, but rather an imagined one of infinite size with the Earth at its center. The celestial equator corresponds exactly with the Earth's equator, and is essentially a projection of the Earth's equator onto the imaginary celestial sphere.
The idea of an imaginary line on an imaginary sphere of infinite size in the sky can be difficult to grasp for many people. For this reason, representations of the celestial sphere and the celestial equator are often depicted in teaching materials as having a set size, somewhat larger than the Earth, simply for the purpose of illustration. This creates the perspective of a viewer of looking down on the sphere, with the Earth at its center. On the sphere, the celestial equator occupies the corresponding path and position of the Earth's equator.
The Earth's rotation causes the celestial sphere to appear to move, but this is not the case. It is fixed in place, but the observer moves as the Earth's rotates. This causes the sphere to appear to rotate once every 24 hours, the same time as the Earth takes to complete one rotation, or one day. The celestial equator, however, will always appear in exactly the same point from any single place on earth. This is because the Earth's equator does not move in relation to any point on its surface, and as the celestial equator exactly matches the Earth's equator it does not move, either.
The position of the celestial equator in the sky will be different, however, depending on the observer's location on the Earth's surface. It will always follow a path describing a straight line joining two points, due east and due west. When standing at the equator, it will pass directly overhead. As the observer moves northwards, the center point of the line moves to the south. Conversely, if the observer moves south of the equator, the path of the celestial equator will appear to move to the north. From the perspective of someone standing at either of the two poles, the celestial equator exactly follows the horizon.
The coordinate system based on the celestial sphere is used both to locate and to quantify the position of objects in the sky. The coordinates for this system are based on declination and right ascension. Declension refers to an object's position above or below the celestial equator and is measured in degrees. Right ascension roughly corresponds to longitude, but uses a specific point on the celestial equator called the vernal equinox as a reference point. Right ascension is measured in hours, reflecting the apparent rotation of the sphere in relation to the Earth.
