Koszul complex is important for homological theory of commutative rings. However, it's hard to guess where it came from. What was the motivation for Koszul complex?
Motivation for Koszul complex
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commutative-algebra
homological-algebra
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2In addition to the answer below. If you know about the Koszul dual of a Koszul algebra $A$, then it is isomorphic to the Ext-algebra $Ext_A(k,k)$ where $k$ is the (direct sum of distinct isoclasses of) simple module(s) of $A$. The natural computation of Ext-groups come from looking at complex, the Koszul complex is exactly (quasi-isomprhic to) the projective resolution of the simple module(s). Beilinson-Ginzburg-Soergel's paper is the place to look at this approach. – 2012-06-28
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1There is a related question here http://mathoverflow.net/questions/146353/history-of-koszul-complex, along with some interesting answers. – 2014-04-17