The Cafe Ellipse at Schiphol Airport, Amsterdam
Introductory Astronomy: Ellipses
(And you'd better not confuse ellipses with eclipses!)
Kepler's first law is that planets orbit on ellipses with the sun
at one focus. An ellipse is (sort of) an oval shape, with two
interior points called foci (singular: focus), a long axis (the major
axis), a short axis (the minor axis), and a center (which should
under no circumstances be confused with a focus). Half of the major
axis is called the semimajor axis, and the semimajor axis
is also the average sunplanet distance. Kepler's first law also
works for other twobody situations where one body outweighs the
other by a large factor, like (1) the earthmoon system, (2) the
JupiterJupiter's moons system, and (3) any sunplanet, suncomet,
sunasteroid system.
Ellipses are a class of mathematical shapes. The circle is the
special case of the ellipse that happens when the two foci (and the
center) are coincident. The number that characterizes how flat the
ellipse looks is called the eccentricity, denoted by the letter e.
The eccentricity e can be calculated by taking the
centertofocus distance and dividing it by the semimajor axis
distance. The limiting cases are the circle (e=0) and a line segment
line (e=1). Below is a picture of what ellipses of differing
eccentricities look like.
 Important ellipse numbers:

a = the length of the semimajor axis

b = the length of the semiminor axis

e = the eccentricity of the ellipse. e^{2} = 1  b^{2}/a^{2}.

Important ellipse facts:

The centertofocus distance is ae.

The major axis is 2a.

Perihelion and aphelion (or perigee and apogee if we are talking
about earth) are the nearest and farthest points on the orbit. These
points are on the major axis, as are both foci and the
center. The perihelion distance is a(1e) and the aphelion
distance is a(1+e). For example, the Earth's eccentricity is
0.017 and its semimajor axis is 1.000 A.U., so its perihelion
distance is (10.017)a, or 0.983 A.U. Similarly, Earth's aphelion distance
is 1.017 A.U.