Ellipse
An ellipse is like a circle that can be squashed.
A circle has one center, but an ellipse has two foci ("A" and "B" below).
Definition
An ellipse is the set of all points on a plane whose distance
from two fixed points add up to a constant.
So, no matter where you are on the ellipse, you can add up the distance to point "A" and to point "B" and it will always be the same result.
(The points "A" and "B" are called the foci of the ellipse) 

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).
Therefore, the ellipse is a conic section (a section of a cone). 
Calculations

Area
The area of an ellipse is π × r × s
(If it is a circle, then r and s are equal, 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!
