  For any affine-variety code we show how to construct an ideal whose solutions
correspond to codewords with any assigned weight. We classify completely the
intersections of the Hermitian curve with lines and parabolas (in the
$\mathbb{F}_{q^2}$ affine plane). Starting from both results, we are able to
obtain geometric characterizations for small-weight codewords for some families
of Hermitian codes over any $\mathbb{F}_{q^2}$. From the geometric
characterization, we obtain explicit formulae. In particular, we determine the
number of minimum-weight codewords for all Hermitian codes with $d\leq q$ and
all second-weight codewords for distance-$3,4$ codes.
