If you think of an ellipse as a 'squashed' circle, the eccentricity of the ellipse gives a measure of just how 'squashed' it is.
It is found by a formula that uses two measures of the ellipse.
c is the distance from the center to a focus.
a is the distance from that focus to a vertex
The formula produces a number in the range 0..1 If the eccentricity is zero, it is not squashed at all and so remains a circle. If it is 1, it is completely squashed and looks like a line. In the applet above, drag the orange dots to create both these eccentricities and some in between.
The word means "off center". It is probably used because the more eccentric an ellipse is, the more its foci are 'off the center' of the ellipse. Kepler discovered in the 1500's that planets are often in "eccentric orbits" instead of exact circles. These orbits turned out to be ellipses with the sun at one of the focus points. In this context, the eccentricity of the ellipse indicates how far from circular these orbits are.
Many textbooks define eccentricity as how 'round' the ellipse is. Since the value increases as the ellipse is more "squashed", this seems backwards. For that reason it is described here as how out of round,or squashed, it is.