Applications of Gröbner bases
Since Grobner basis algorithms may be considered as nonlinear generalizations of Gaussian elimination for systems of linear equations, they have very widespread applicability. Below is a random collection of applications of Grobner bases.
effective computation with (holonomic) special functions
solving Diophantine equations (Pell)
automated geometry theorem proving.
coding theory
signal and image processing
robotics
graph coloring problems e.g. Sudoku puzzles
extrapolating "missing links" in palaeontology, and phylogenetic tree construction
find intersection points of a couple of conics (pick the right coefficients to make it not so tedious to do all the manipulation)
describing the motion of a constrained single hinged robot arm or planetary epicycles (make a cardioid from two equations)
colorability of a graph (see A Crash Course... ) (when presented with the construction, very easy to see that the algorithm produces a solution)
Here are the things I use Grobner bases for, which I certainly find interesting:
Extending the univariate division algorithm to multivariate polynomials (although not a true euclidean division algorithm, it is still useful).
(related) Computing generators for $I_1 + I_2$ where $I_1,I_2$ are ideals in a multivariate polynomial ring (say $\mathbb{C}$), and using this to determine $I(V_1\cap V_2)$ where $V_1$ and $V_2$ are affine varieties in $\mathbb{A}^n$ for $n > 1$.
I'm not sure if these interest you or the students you are presenting to, but hopefully it's at least a start.