On an (x, y) graph an ellipse can have the shape shown to the left.
Here's a link to EZ Graph - Ellipse. This is a graphic calculator that presents functions that draw an ellipse.
If a plane intersects the cone all of the way around but is not perpendicular to the axis, the conic section is an ellipse. This is shown to the left.
Click this link for an interactive 3D virtual reality world showing the ellipse as the intersection of a cone and a plane.
Here is a video that shows several possibilities for forming an ellipse by intersecting a cone and a plane.