A satellite in a direct orbit with an orbital period less than one day will move from west to east. This is called apparent direct motion. A satellite in a direct orbit with an orbital period greater than one day will move from east to west, in what is called "apparent retrograde" motion. This is because the satellite is orbiting in the same direction as the Earth's rotation, but more slowly than the Earth. Any satellite in a retrograde orbit will move from east to west.
As the orbital period of a satellite increases, approaching the rotational period of the Earth, its sinusoidal ground track will become compressed laterally, meaning that the points at which it crosses the equator will become closer together.
A satellite whose orbital period is exactly the same as that of the Earth is said to be in a geosynchronous orbit. Its ground track will have a "figure eight" shape in a fixed location on the Earth, crossing the equator twice each day. It will move in the prograde direction when it is on the part of its orbit closest to perigee, and in the retrograde direction when it is closest to apogee.
A special case of the geosynchronous orbit, the geostationary orbit, has an eccentrity of zero (meaning the orbit is circular), and an inclination of zero in the Earth-Centered, Earth-Fixed coordinate system (meaning the orbital plane is not tilted relative to the Earth's equator). The "ground track" in this case consists of a single point on the Earth's equator, above which the satellite sits at all times. Note that the satellite is still orbiting the Earth — its apparent lack of motion is because the Earth is rotating about its own center of mass at the same rate as the satellite.
If the argument of perigee is zero, meaning that perigee occurs on the equatorial plane, the ground track of the satellite will appear the same above and below the equator. However, if the argument of perigee is non-zero, the satellite will behave differently in the northern and southern hemispheres. The Molniya orbit, with an argument of perigee near 90°, is an example of such a case. In a Molniya orbit, apogee occurs at a high latitude (63°), and the orbit is highly eccentric (e = 0.72). This causes the satellite to "hover" over a region of the northern hemisphere for a long time, while spending very little time over the southern hemisphere. This phenomenon is known as "apogee dwell", and is desirable for communications for high latitude regions.