The Horizon Coordinate system is defined with respect to an individual observer standing on the earth. The horizon in astronomy is not quite the same as the horizon we see on earth. It is idealized to be free from geographical effects such as mountains, valleys, or even the bending of the earth. It is a flat plane tangent to the earth's surface where the observer is standing. Consequently, an observer can only see half of the celestial sphere at any given moment.
There are two defining points/regions for the horizon coordinate system. The horizon plane is one where direction are given with respect to the cardinal points: north, east, south, and west. The other point is straight up and is called the zenith.
Azimuth is the coordinate defining directions parallel to the horizon (red in figure to the right). Azimuth goes from 0 to 360° starting with north = 0° and increasing towards the east. That is, east is azimuth 90°, south is azimuth 180°, and west azimuth 270°. Stars with the same azimuth lie on an arc from the zenith through the object down to the horizon (meeting perpendicularly) called a vertical circle (though technically it's only half a circle).
Because the earth rotates around an axis aligned with the earth's poles, the sky moves east and west, but not north and south. The observer's meridian arc going from from the north point of the horizon, up through the zenith, and down to the south point of the horizon. The meridian is an important reference as it is the place where an object in the sky (like the stars, sun, or moon) will be “highest” in the sky.
Altitude is the coordinate defining directions above or below the horizon plane (blue in figure to the right) – how high an object is in the sky. It measures the position of a star on a particular vertical circle. The horizon is altitude = 0°. Straight up – the zenith point – is altitude 90° and straight down, below the horizon is the nadir at altitude -90°