Ellipse (plane making an angle with the base less than the angle of the cone's side).
Parabola (plane parallel to the side of the cone)
Hyperbola (plane's angle with the base is greater than that of the side).

Of course, this assumes that you do not run through the apex of the cone. If you do that, you get the degenerate cases of a point, line, and crossed lines instead of ellipse, parabola, and hyperbola respectively.