Question on Polynomials in Finite Fields

Question

Suppose we have a finite field $\mathbb{F}^n_q$ and a polynomial $P\in Poly_D(\mathbb{F}^n)$ such that $P$ vanishes at every point of $S\subset{\mathbb{F}^n_q}$. I'm not sure what it means for a polynomial to vanish at every single point in a subset of a field, and how this can even be possible.