An ellipse is a stretched circle. Slice a cone at a slant and an ellipse falls out. Planets orbit on ellipses, racetracks are roughly elliptical, and whisper galleries are ellipsoidal rooms where a quiet word at one focus carries to the other.
e = 0 → circle · 0<e<1 → ellipse · e = 1 → parabola · e > 1 → hyperbola
The shaded wedge is the area swept by the radius in a fixed slice of time — it stays the same size all the way round (Kepler's 2nd law), so the planet must speed up when it's close to the Sun and crawl when it's far. e = 0 ⇒ a circle; bigger e ⇒ a more stretched ellipse with the Sun further off-centre.
a = semi-major axis (long), b = semi-minor axis (short)
Sum of distances from any point P to the two foci is constant — equals the long axis length.
Kepler's first law: every planet moves in an ellipse with the Sun at one focus. The other focus is just an empty point in space.