Jordan canonical form has very interesting properties and profound implications. For example, if the Jordan canonical form $J$ is diagonal, then it is equivalent to say that it corresponds to a matrix $A$ having distinct eigenvalues the diagonal elements of $J$. This is helpful in quickly testing the diagonalizability of matrices by inspection.
The Jordan matrices for non-diagonalizable matrices are more interesting. For me it is not well understood what each 1s on the upper diagonal means. It is also not clear how one can utilize this matrix some how aside from proving that $A$ is not diagonalizable.
Can someone point out some interesting properties of Jordan forms that I might not realize but is useful in studying linear systems (whatever meaning of system)?