Polynomial method for complexity results
Some classic examples of the use of the polynomial method are:


Razborov-Smolensky's proof of "Parity not in $AC^0$" (lots of expositions available online, here is one)
Beigel-Reingold-Spielman's proof of "PP is closed under intersection"
Bazzi's result for fooling DNFs, and Braverman's proof of "Polylog independence fools $AC^0$"


Also, fourier analysis of boolean functions (here is a great course by Ryan O'Donnell)  has a HUGE collection of awesome results, my favourite being the Kushilevitz-Mansour-Nisan's proof of the Goldreich-Levin theorem. 

Scott Aaronson had in fact given a tutorial at FOCS'08 on the "The Polynomial Method in Classical and Quantum Computing (ppt)". 

Hope this helps.