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Kepler's First Law of Planetary Motion says that all orbits of planets in the Solar System are ellipses, and the Sun is at one of the foci of each planet's elliptical orbit. We further extend this law to hold true for any celestial body going around any other celestial body on a repetitive basis, with the center of mass of that two body system at one focus.
An ellipse is an oval, typically sort of a squished circle. One definition of an ellipse is any section, or cut, of a cone, that is closed. The picture to the left depicts the general form of such a cut. If the cut is perpendicular to the axis of symmetry of the cone, it becomes a circle (all circles are ellipses, but not all ellipses are circles).
Another definition of an ellipse is the set of all points such that the sum of the distances of any of those points to two other fixed points (called foci, the plural of focus) is a constant.
Half the length of the major axis is called the semimajor axis, and is typically indicated by the letter a in any formula about ellipses. Similarly, half the length of the minor axis is called the semiminor axis, and is typically indicated by the letter b. The distance from the symmetrical center of an ellipse to either focus is called the focal length, and is typically indicated by the letter c. The farther apart the two foci of an ellipse (the bigger is the value 2c), the flatter or more squished is the ellipse. We quantify this "squishedness" by the variable e, the eccentricity, where e = c/a. (The eccentricity of a circle is zero, since c = 0.)
The current eccentricity of the Earth's orbit is low, or "mild", only .0167 (1.67%). The Moon's average orbital eccentricity is .0549 (5.49%). Among the eight planets of the Solar System, the planet Mercury has the highest orbital eccentricity, currently at .2056 (20.56%).
By comparison, the dwarf planet Pluto's current orbital eccentricity is .248 (24.8%), and Halley's Comet (which comes by once every 76 years) has an orbital eccentricity of .967 (96.7%).
While it is correct to say that the Sun is on one of each planet's orbital foci, it is not correct to say that all the planets have the same foci. Where the other focus of a planet's orbit is depends on that planet's orbital eccentricity. See here for further details.
Click here to play with an ellipse generator. You can leave the value of a at 10, but dial in the eccentricities for Earth, the Moon, Mercury, Pluto and Halley's Comet, to see how the shape the corresponding ellipse changes (becomes more squished). For the Moon, check that e = c/a.
Consider now any straight line segment that goes through the symmetrical center of an ellipse and terminates on two points of the ellipse. For a circle, such a line segment would be called a diameter. For an ellipse which is not a circle, the longest such line segment is called the major axis, and the shortest such line segment is called the minor axis.
For Earth, currently, the solstice days occur very near the days when Earth is on the major axis endpoints, and the equinox days occur very near the days when the Earth is on the minor axis endpoints.
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