An ellipse is a squashed circle with two focus points or foci, planets orbit in an elliptical path. On the diagram to the right the Sun sits at one of the foci, and the other foci is empty (black dot), the planet orbits around the ellipse.
The amount the ellipse is squashed, or the 'flattening' is called the eccentricity. The more squashed, the higher the eccentricity. A circle has an eccentricity of 0, the more squashed an ellipse becomes more flattened the eccentricity approaches 1. So, all ellipses have eccentricities lying between 0 and 1.
Try using the electric orrery to see how eccentric these orbits are.
The longest axis of the ellipse is called the major axis, while the short axis is called the minor axis. Half of the major axis is termed a semi-major axis, and likewise half of the minor axis is called the semi-minor axis.
There are 3 laws which describe the motion of planets around stars, known as Kepler's laws, which are described in this European Space Agency (ESA) video.