The polynomial method is a powerful, albeit somewhat mysterious and fragile, tool in extremal combinatorics, being used for instance in Dvir's proof of the Kakeya conjecture over finite fields:
http://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/
This is currently the only known proof of the full conjecture. Previously to Dvir's work, algebraic geometry methods did not feature prominently in the prior partial results.
A related method is Stepanov's method to count points in algebraic varieties over finite fields, though this is clearly a question which was already well within the purview of algebraic geometry to begin with.