Ellipse
An ellipse is like a squashed circle.
A circle has one center, but an ellipse has two foci ("F" and "G" below).

If you go from point "F" to any point on the ellipse and then go on to point "G", you will always travel the same distance.
f+g is always the same
Definition
An ellipse is the set of all points on a plane whose distance
from two fixed points F and G add up to a constant.
The points "F" and "G" are called the foci of the ellipse (F is a focus, G is a focus, and together they are two foci) 

Draw It
Put two nails in a board, put a loop of string around them, and insert a pencil into the loop. Keep the string stretched so it forms a triangle, and draw a line ... you will draw an ellipse.
A Circle is an Ellipse
In fact a Circle is an Ellipse, where both foci are at the same point (the center). In other words, a circle is a "special case" of an ellipse.

Section of a Cone
You can also get an ellipse when you slice through a cone (but not too steep a slice, or you get a parabola or hyperbola).
In fact the ellipse is a conic section (a section of a cone) with an eccentricity between 0 and 1. 
Calculations

Area
The area of an ellipse is π × a × b
(If it is a circle, then a and b are equal to the radius, and you get π × r × r = πr^{2}, which is right!) 
Perimeter Approximation
Rather strangely, the perimeter of an ellipse is very difficult to calculate, so I created a special page for the subject: read Perimeter of an Ellipse for more details.
But a simple approximation that is within about 5% of the true value (so long as r is not more than 3 times longer than s) is as follows
Remember, this is only a rough approximation!

Equation
By placing an ellipse on an xy graph (with its major axis on the xaxis and minor axis on the yaxis), the equation of the curve is:
x^{2}/a^{2} + y^{2}/b^{2} = 1
(very similar to the equation of the hyperbola: x^{2}/a^{2}  y^{2}/b^{2} = 1, except for a "+" instead of a "") 
