Courses about linear algebra make great demands on looking for eigenvalues and transforming matrices to diagonal matrices (or, at least, to Jordan normal form). This is somehow a technical, recipe-like part where many students don't see a deeper understanding.
What is "the best" showcase why one needs eigenvalues? Or: What is a good motivation to come up with where they will be needed (should be understandable for the student at that point)?
Obtaining formulas for the $n$-th term in a linear recurrence, such as Fibonacci numbers, is one application that certainly does not overtly mention linear algebra in the set-up.
I like Markov chains and Google PageRank (which is essentially a special kind of Markov chain). It doesn't take very long to explain and motivate Markov chains and to argue that the probability distribution at time $n$ is the $n$'th power of the transition matrix times the distribution at time $0$. You can then start talking about how to calculate powers of matrices. You can draw pictures of Markov chains, which is more pleasant than just writing down matrices. You can make a compelling argument for the usefulness of very large matrices, which can seem gratuitous in other contexts. You can draw Markov chains that have repeated eigenvalues for obvious reasons of symmetry, which is instructive. You can show that $1$ is always an eigenvalue, using an argument that brings together various other ideas in an interesting way. You can make heuristic arguments about long term behaviour, and justify them algebraically with reference to eigenvectors of eigenvalue $1$. It is not hard to convince the students that they should care about PageRank.
My notes covering this are at http://neil-strickland.staff.shef.ac.uk/courses/MAS201/
Imagine a linear mapping $f: R^2 \to R^2, e_1 \mapsto (1.5, 0.5), e_2 \mapsto (0.5, 1.5)$. (As long as $R$ contains the numbers $1.5$ and $0.5$, it could be any ring. The real numbers serve as the most convenient example, however.)
Can we "see", what the mapping does? Can we "see" what $f^5$ does? Given a basis of the two eigenvectors, $(1,-1), (1,1)$ we find the first one has eigenvalue 1, the second one eigenvalue 2. Take the time to sketch $e_1, e_2$ and their images on the blackboard. Students will hardly see any structure. Now, take the basis of eigenvectors and sketch their images. Students will immediateley "see" the mapping (hopefully). They then can describe $f^5$ without calculations.
As one application you might take the fibonacci numbers. Their growth can be described using matrix multiplication (see wikipedia) and determining the eigenvector means determining an explicit formula for the implicitly defined series.
I have always found the standard motivations for eigenvalues to be a little artificial. The primary application for eigenvalues is ultimately diagonalization and there are several ways you could try to motivate diagonalization:
Taking large powers of matrices seems to be a popular one. But its not immediately obvious what this is used for.
An extension of taking large powers using diagonalization is using it to find solutions to linear recurrences. But chances are you have never talked about linear recurrences before this point. Introducing a concept and then a problem that the concept solves is a little backwards. The problem should come before the solution.
Steven Gubkin's suggestion of using it for the definiteness of a matrix is also a good one. But again, definiteness is often not part of a first course in linear algebra and many students study linear algebra before they do multivariable calculus where they need to test the definiteness of the Hessian.
Using it to solve differential equations is yet another example, and yet again, many students may not have seen differential equations yet.
My point is, all of the standard attempts to introduce diagonalization are application based and such attempts are heavily dependent on audience. Worse yet, when improperly executed the examples appear contrived.
Diagonalization is ultimately a structure theorem. Anyone can appreciate the fact a diagonal matrix is much easier to study than a non-diagonal matrix. Diagonalization then offers a power tool to study any property which is invariant through similarity by reducing a matrix which we know little about to a matrix which is zero everywhere except the diagonal. That is motivation enough in my experience, and you could always fall back onto one of the standard motivating examples if your students are still not convinced.
For a real showcase, I recommend a scenario where resonance frequencies play a role.
Suspension bridges are real-world objects which are delicate enough that soldiers are usually not allowed to march over bridges (the German traffic law StVO states this in § 27 Abs. 6). The reason for this is that small, regular excitations with a certain frequency will cause the bridge to oscillate substantially. This can be seen in this video of the Millennium bridge in London.
In order to calculate these so-called resonance frequencies, one can try to describe such a bridge with the help of a differential equation; in the easiest case as a vibrating string. When you solve the corresponding differential equation, you will get a matrix whose eigenvalues are exactly the resonance frequencies.
Depending on the knowledge on differential equations one may or may not start to work with these.
Note that the Galloping Gertie did not collapse due to resonance, but because of Aeroelastic flutter.
If you don't like bridges, playing with electrical circuits might also provide nice visible or audible results.
One of the standard examples is from stability analysis of dynamical systems. In the linear approximation you ask whether the $0$ solution is stable for $$ X_{j} = A X_{j-1} $$ (discrete time) or $$ \frac{d}{dt} X = B X $$ (continuous time).
Another cool application is to figuring out whether a given matrix is positive definite. This is useful for the second derivative test, for example.
Suppose for simplicity that the matrix $A$ of some linear mapping $\mathbb{R}^n \to \mathbb{R}^n$ is symmetric (hence diagonalizable) and all the eigenvalues $\lambda_1,\ldots,\lambda_n$ are different. Then, there is an orthogonal base such that in that base
$$\text{$A$ behaves like scaling with factors $\lambda_1,\ldots,\lambda_n$ respectively.}$$
In other words, the eigenvalues tell us how the mapping looks like, is it a bit like a symmetry (e.g. when $\lambda_i < 0$), or perhaps like a projection (e.g. when $\lambda_i = 0$) or maybe a contraction (e.g. $\lambda < 1$), etc.
This is kind of similar to the decomposition theorem for abelian groups, i.e. it lets you simplify the mapping into one which can be more easily handled, e.g. has no weird cross-dependencies and each part is by itself well understood. No wonder it is so important!
I hope this helps $\ddot\smile$
One clear motivation is the relationship to optimization: The largest eigenvalue of a real symmetric matrix is the maximum value of the (normalized) quadratic form $$ \frac{x^{\mathsf T} A x}{x^{\mathsf T} x} $$ This is used for instance in spectral graph theory and elsewhere.
Markov chains have already been mentioned, that is a nice direction.
Of course, zero is an eigenvalue iff $A$ is singular. I'm not sure I'm creative enough to take this as a start point for motivating the eigenvector concept.
Another way to discover them is simply by taking a point $x_o$ and a matrix $A$ and calculating $x_1=Ax_o$,... $x_k = Ax_{k-1}$. Since we're looking for a visualization, you want to use a $2 \times 2$ matrix. You'll find that the points are drawn into the eigen-axes or, if it happens to have complex e-value it spirals. This requires not too much curiousity and it immediately reveals real and complex e-values existence for real $2 \times 2$ matrices. See page 221-223 of http://www.supermath.info/math321v2.pdf for my not so pretty attempt at this discussion/discovery section.
I'm editing those notes in the past week or so when it hit me, it would also be good to emphasize the simple geometric example of the axis-vector of a rotation. Perhaps it is interesting that the axis may reside in the space $n=3$ or may not $n=2$. Again, the necessity of both real and complex eigenvalues is revealed by a topic which is a natural curiousity (rotations in $n$-dimensions ).
All of this said, the application of Jordan chains to unravel the matrix exponential and assemble the general solution to $x'=Ax$ is to my taste a powerful showcase. However, the intended audience needs some desire and/or familiarity with ODEs to appreciate it. While we're at it; I have found the connection between generalized eigenvectors and the double-root solution to be a satisfying justification for studying them in years past. I mean, I want to be able to solve all the ODEs with constant coefficients, not just the problems which happen to be diagonalizable. I wish more of us thought generalized e-vectors were worth teaching.
If you have a physicsy audience it's easier. Eigenvectors are the observable states. Eigenvalues are the values of energy, momentum, spin etc... of the system which the states represent. That's fundamental physics. The question of simultaneous diagonalization becomes interesting as it is the question of being able to simultaneous know the values of a pair of observables for a given pair of physical quantities. For a non-example, momentum and position $[x,p] \neq 0$ hence we cannot find simultaneous eigenstates of $x$ and $p$. This leads to the famous Heisenberg Uncertainty Principle.
Another way:
You can introduce students to the recurrence relation: $$ a_{n + 1} = a_n + 2b_n $$ $$ b_{n + 1} = a_n + b_n $$
And show that this recurrent relation can be represented as a matrix multiplication:
$$ v_{n+1} = Av_n $$
where $ A = \binom{1 \ \ 2}{1 \ \ 1}$ and $ v_n = \binom{a_n}{b_n} $. Then show (just demonstrate, not prove) that $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \sqrt{2}$. Then you can do a visual representation that shows that the matrix $A$ is a skew (and reflection, I guess) matrix.
Using eigenvalue and eigenvectors, you can show convergence to $\sqrt{2}$, and (if you put some effort into it) make similar recurrent relations for other squareroots (and argue about their convergence: we want the eigenvector that it needs to converge to to have big eigenvalue and the other eigenvector a small eigenvalue). Also, if there is a basis of eigenvectors (in this case, there is), you can give an explicit formula for each point in the recurrent relation which is kinda cool.
You can motivate eigenvalues as one way to analyze this kind of recurrent relations.
This is not the easiest way to handle recurrent relations, but it gives students a geometrical example, and shows an application at the same time. I think it is also good to present this idea that matrix multiplication can be used to describe recurrent relations.
This may be too advanced for some students in a first-semester linear algebra course, but those who have had some Physics may be impressed by the diagonalization of a moment of inertia tensor. Take some irregularly-shaped 3-dimensional object, and form a $3 \times 3$ matrix describing all of the components of the moments of inertia around various axes; the equations of motion for rotating the object around an arbitrary axis are extremely complicated. Now diagonalize the system, finding the principal axes of the object: suddenly they are simple (if you rotate around one of those axes).
Demonstration: Bring in a rock or some other irregularly-shaped object and spin it like a top. We instinctively align it with one of its principal axes, because we can "feel" them in our hands.
One very good application of Eigen Value and Vectors is Principal Component Analysis in Machine Learning. This application can actually help to visualise the mathematical concept better.
