Because the Earth rotates on its axis, all objects appear to travel across the sky once every day. The orientation of the circles as viewed by observers is determined on their Earth's latitude. Depending on where you are, certain stars are visible all the time, while others rise and set and some never appear. The brightest stars in the night sky are those that never disappear from view during the course of a year.
At the equator, the circle of sight is vertical, so if something there is rising on the eastern horizon it will be setting on the western one, and vice versa. At the poles, all points on the rotating sphere are seen over a horizontal circle, so nothing ever disappears below the horizon.
But even at the poles, there are stars that come close enough to the horizon to dip beneath it. These include the polar stars, which are very bright stars that always lie just above the horizon. They can only be seen from within certain regions, such as parts of Antarctica where they serve as reliable navigation lights. Otherwise, they would not be able to be seen from anywhere on Earth.
Beyond the poles, the horizon is actually curved. So even though everything is seen over a horizontal plane, there are portions of the circle of vision that go beyond this plane - including some areas near the center of the globe - where things are seen sinking below the horizon.
The apparent motion of a star to an observer caused by the Earth's rotation depends on the observer's position on Earth as well as the star's location relative to the Earth's rotation axis. If the star is located near the rotation axis, it will appear to move more quickly from north to south than if it were far from the axis.
Stars that lie close to the North Pole appear to move faster than those closer to the South Pole. This is because both poles are near the rotation axis and thus appear nearer to the center of the Earth during certain times in their orbits. When viewed from above the Arctic Circle, for example, the stars seem to rush by because the North Star keeps appearing in a new part of its arc across the sky.
Stars that lie far from the rotation axis do not experience a rapid movement across the sky. They appear to stay put while the rest of the universe moves past them. Stars that lie close to the equator appear to move slowly through the night because they are near the center of the Earth and don't experience any acceleration due to gravity.
The direction of stellar movement changes as the Earth rotates. So instead of moving across the sky, stars appear to circle around a point somewhere over the horizon.
The Sun, Moon, and planets appear to travel through the sky in the same way as the stars do. Because of the Earth's rotation, everything in the sky appears to move in unison, spinning once every 24 hours. Even though the Sun, Moon, and planets are fixed in place, they seem to move across the sky because we on Earth are moving too.
Stars don't emit light themselves; they just reflect light from other distant objects such as galaxies or other stars. So stars don't move with respect to each other. Instead, it is the background objects that move with respect to them. Because our planet is moving though, all these background objects appear to move along with us.
For example, when you look up at the night sky and see the Moon, it is actually moving away from us. But since we're on Earth, it looks like the Moon is standing still while the stars around it are rushing by.
This effect explains why objects in the sky look as if they're moving even though they're not. And since stars don't move, this also means that the Earth's location in space isn't important for seeing stars disappear over the horizon or reappear after being gone for a while. What matters is where the Earth happens to be at any given moment.