An ellipse is a shape defined by two focal points and a taut string. Drawing an ellipse involves placing two points on a page and using a string and pencil to trace the shape while keeping the string taut. Ellipses have the property that light rays or sound waves bouncing off the curve will reflect back to the two focal points, explaining whispering galleries where people standing at the focal points can hear each other clearly. The document provides examples of writing the equation of ellipses given properties like the center, vertices, or axes.