Take two cones and connect them at their apexes (the "point".)
Now throught this figure we pass a plane.
(1) If the plane is perpendicular to the axis of the cones, and intersects one of the cones in more than one point it cuts a circle.
(2) If the plane is oblique to the axis of the cones and cuts through one of the cones it cuts an ellipse.
(3) If the plane is oblique to the axis and only cuts one cone, but does not cut all the way through it cuts a parabola.
(4) If the plane cuts both cones it cuts a hyperbola.
Thus the conic sections are the circle, ellipse, parabola, and hyperbola.
** If the plane cuts only the point where the cones meet it cuts a point. If the plane runs along the edge of the cones it cuts a line -- thus points and lines are sometimes considered as degenerate conics.
See the link for pictures.