What does Canonical Form Means Intuitively?

Question

I have seen multiple times where canonical form is mentioned. I went to Wikipedia and as usual its quite formal definition and not intuitive at all. So the following context taken from MathWorld, what exactly is canonical form?

Matrix diagonalization is equivalent to transforming the underlying system of equations into a special set of coordinate axes in which the matrix takes this canonical form.

Answer 1

For me "intuitive" means "graphical" or "visual."

Here is an ellipsoid representing a general system of quadratic equations:

Here is the visual representation of that system of equations in "canonical form" (through diagonalization):