What is the eccentricity of an ellipse and what does it tell me?

Eccentricity e measures how stretched or flattened a conic is. For an ellipse with the standard equation x²/a² + y²/b² = 1 (with a greater than b), the eccentricity is e = root of (1 − b²/a²), and it always lies strictly between 0 and 1. When e is close to 0 the ellipse is almost a circle, because the two axes are nearly equal. As e gets closer to 1 the ellipse becomes long and flattened. A circle is the special case e = 0. The foci sit at a distance ae from the centre along the major axis, so a larger e pushes the foci further out. The same b² = a²(1 − e²) relation lets you find b once you know a and e.