Let's first look at a typical orbit first for comparison. Note that the object being orbited could be anywhere.
Note that it spends a time both above and below the equator. A few things about a Geosynchronous orbit:
- The inclination varies with time, although this isn't that important for day to day operations.
- The period is 24 hours, but a geostationary satellite either isn't perfectly circular, or 0 inclined.
The orbit you have shown is an elliptical inclined geosynchronous orbit. The shape is actually called an Analemma. In fact, the image you provided is of the Sun doing something similar to what these spacecraft will do on a daily basis.
The inclination is responsible for the up and down motion. The sideways part is due to the eccentricity primarily, although an inclined orbit will show a slight variation due to the changing relative velocity compared to the Earth's surface the higher it goes. The spacecraft moves faster when closer to Earth, and thus will be somewhat ahead of the Earth. When it is further, it moves slower, moving back in the other direction.