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Hecke operators are things that act on modular forms and give rise to a lot of interesting arithmetical results:

http://en.wikipedia.org/wiki/Hecke_operator

On the other hand on the wikpedia page for Hecke algebras, which should naively be an algebra of Hecke operators, the term seems to acquire very different meanings, such as deformations of group algebra of a Coxeter group.

What is the connection between the two?

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    There are at least two MO questions about this: http://mathoverflow.net/questions/19684/relation-between-hecke-operator-and-hecke-algebra and http://mathoverflow.net/questions/4547/definitions-of-hecke-algebras .2010-11-29
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    The connection is also explained at the Wikipedia article on Hecke algebras: http://en.wikipedia.org/wiki/Hecke_algebra#Hecke_algebra_of_a_locally_compact_group2010-11-29

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