What is the purpose of Jordan Canonical Form?
Two square matrices $A$ and $B$ are said to be similar, or conjugate, if there exists an invertible square matrix $P$ such that $A = P^{-1}BP$. This is equivalent to saying that $A$ and $B$ represent the same linear transformation in different bases, with $P$ providing the change-of-basis matrix that relates them.
If one wants to solve a linear equation but is working in an inconvenient basis, it may help to change the basis to a more convenient one. Sometimes one can find a convenient basis by inspection, but in general one often changes the basis to obtain the Jordan canonical form of the desired matrix. For solving linear equations the Jordan canonical form is ideal, since (1) it has a very simple structure (upper triangular, and only $1$-s just above the diagonal) and (2) it can be computed for any square matrix.
It is important for theoretical reasons to know that one can always find the Jordan canonical form of a square matrix. It simplifies many abstract proofs to assume a matrix in the proof is in Jordan canonical form. If you know a little abstract algebra, the Jordan canonical form is also of interest in the sense that it completely classifies the conjugacy classes of matrices over the complex numbers (and some other fields as well), and is a special case of a more general phenomenon regarding module homomorphisms.
However, as for more real-world purposes the Jordan canonical form is not ideal. The primary example of a real-world application would be solving a system of linear equations (for example, one that comes up when trying to solve a system of linear ODEs), and unfortunately the Jordan canonical form is not well-suited to this task in practice. The reason is that the Jordan canonical form is very sensitive to perturbations in the original matrix; that is, if an entry $a_{ij}$ in the matrix $A$ is perturbed to $a_{ij}+\epsilon$, it is very possible for the Jordan canonical form of the new matrix to be wildly different from the original Jordan canonical form. (That is, the Jordan canonical form is not numerically stable.)
The numerical instability of the Jordan canonical form makes it bad in real-life applications, where systems of linear equations arise from real-world data that always has a level of uncertainty. For this reason, in real-world applications one must abandon the Jordan canonical form for numerically stable algorithms. One example of such an algorithm is the Schur factorization, which also transforms (using unitary matrices) a matrix into a conjugate upper triangular matrix, and thus simplifies the solution of linear systems.
Among other things the Jordan form can show that any Markov process must terminate and also helps to find the limiting state. Instructors usually motivate such processes by asking how to find equilibrium states in chemistry, but this also has applications to google's page rank system. Basically rank pages due to the largest entry in the limiting state.