TITLE: Hecke-algebras in your field of mathematics
QUESTION [12 upvotes]: (How) do Hecke-algebras arise naturally in your field of mathematics and why are they important?
How would you define them and how do you think about them? 
e.g. generators and relations, functions on some space, grothendieck group...

REPLY [4 votes]: One very interesting application is the proof by Kuhn and Priddy of the Whitehead Conjecture in stable homotopy theory (MR):

\bib{MR803606}{article}{
   author={Kuhn, Nicholas J.},
   author={Priddy, Stewart B.},
   title={The transfer and Whitehead's conjecture},
   journal={Math. Proc. Cambridge Philos. Soc.},
   volume={98},
   date={1985},
   number={3},
   pages={459--480},
   issn={0305-0041},
   review={\MR{803606 (87g:55030)}},
   doi={10.1017/S0305004100063672},
}

The point of contact is that the proof involves classifying spaces of elementary abelian groups $(\mathbb{Z}/p)^r$ and their suspension spectra.  The $p$-local group ring $\mathbb{Z}_{(p)}[GL_r(\mathbb{Z}/p)]$ acts on this spectrum, and one can use the Steinberg idempotent to give a splitting.  The relationship between the Steinberg idempotent and Hecke operators is pure algebra.
There is also a relationship between Hecke operators and power operations in elliptic cohomology (MR):

\bib{MR1637129}{article}{
   author={Ando, Matthew},
   title={Power operations in elliptic cohomology and representations of
   loop groups},
   journal={Trans. Amer. Math. Soc.},
   volume={352},
   date={2000},
   number={12},
   pages={5619--5666},
   issn={0002-9947},
   review={\MR{1637129 (2001b:55016)}},
   doi={10.1090/S0002-9947-00-02412-0},
}

This actually generalises to give a relationship between power operations in Morava $E$-theory (at the prime $p$ and height $n$) and a kind of Hecke algebra for the monoid of $n\times n$ matrices over $\mathbb{Z}_p$ with nonzero determinant.