Let's start with eigenvectors.
For a linear transformation, there will be vectors whose direction will not be changed by that transformation. Those are eigenvectors.
While the direction of eigenvectors might not change, the magnitude does. The eigenvalue is the magnitude of that change.
What is the relevance of eigenvalues from a pure math point of view? For one, all matrices with the same eigenvalues are similar. Finding the eigenvalues of a matrix allows us to treat our matrix as something similar to a diagonal matrix. And diagonal matrices are really easy to work with.
What are some of the applications?
Quadratic forms -- A quadratic form takes on a few basic shapes, (parabola, hyperbola, eclipse) with higher-dimensional analogs. The eigenvalues tell you what basic shape you have.
System of differential equations -- the solution to the differential equation will often be some combination of trigonometric or exponential functions. The eigenvalues determine those functions.
Principal component analysis -- The eigenvalues show the size/sensitivity of these components.
In a Markov process, a transformation is executed repeatedly. Eigenvalues with absolute values less than one disappear to insignificance. The largest eigenvalue will come to dominate the process.
This is really just scratching the surface, but I hope give you some idea where this line of study is going.