As you recall, two matrices are similar if one is equal to the other after an appropriate change of basis. Similar matrices fall into equivalence classes, and Jordan wanted to understand these classes, and select a canonical representative for each.
As we shall see, any matrix M is similar to a matrix J, where J is a diagonal matrix with some scattered ones along the subdiagonal (just below the main diagonal). Here J is the canonical representative for M, or more precisely, the equivalence class of matrices similar to M. Choose a different coordinate system, and the transformation that was implemented by M is now implemented by J, a much simpler matrix. This works in any field, although you may need to extend the field to its algebraic closure. The new basis, acted upon by J, and the diagonal elements of J, may employ elements from the algebraic closure.
This theorem is almost a corollary when M is diagonalizable. Let J be the diagonal matrix that is similar to M, and J is in jordan canonical form.
When M is not diagonalizable we have more work to do. Let's get started.