A circle can be defined as the shape created when a plane intersects a cone at right angles to the cone's axis. It is one of the four conic sections. (the others are an ellipse, parabola and hyperbola).
In the above figure, there is a plane* that cuts through a cone. A circle is formed at the intersection of the cone and the plane if the plane is at right angles to the vertical axis of the cone (i.e. parallel to the cone's base).
As you drag the plane to the top, the circle gets smaller until it is a single point at the apex of the cone. As you drag it downwards the circle gets bigger. On the left of the applet is a view looking down on the top of the cone, showing the circle's center is on the cone's axis, and varies in size.
* The plane is drawn with edges for clarity, but in reality a plane goes on for ever in all directions.