First, you link to an article about Gaussian Optics which is a specific branch of optics using coherent radiation/light that has nothing to do with conventional photography.

That aside, a lens has a "front" and a "back" so there are two of everything since there is one kind of optical point or optical plane for each side of a lens. In other words, the back can be the front and the front can be the back.

In addition, an imaginary line drawn straight through the centre of a lens is called the optical axis. When you aim a lens, you point the optical axis toward the subject.

Light from a distant point on the optical axis enters the lens and comes to a focus on the optical axis at the rear focal point of the lens which is measured from the surface of the lens in fact.

The light APPEARS to come from a point on the optical axis referred to as the nodal point.

Nodal points and planes are virtual images for the image origin that fall on the optical axis. An image comes to a focus through the lens onto a plane called the image plane (the film or sensor).

The image APPEARS to come from a flat plane within the lens perpendicular to the optical plane called the nodal plane.

Cardinal points lie on the optical axis. Focal points lie where the focussed image is formed. An image is formed on the focal plane located on the optical axis perpendicular to the axis and parallel with the plane of the lens.

Paraxial rays of light enter a lens with angles similar to/within a few degrees of the optical axis. They are "nearly" aligned with the optical axis.

Non-gaussian images are more easily described by the term wave-fronts which are like "slices" of a 3-dimensional image. In gaussian objects, the focal image is more planar.

I hope this helps.