Motivation for Koszul complex 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?

I don't know the historical origins, but it is not so hard to make up a story:
Consider the basic example
$$0 \to k[x] \to k[x] \to k \to 0,$$
where the middle arrow is mult. by $x$.  This is a resolution 
of $k = k[x]/(x)$ as a $k[x]$-module.
Now suppose you want to generalize this to obtain a resolution of $k = k[x_1,...,x_n]/(x_1,...,x_n)$ as a $k[x_1,...,x_n]$-module.  It is not hard to see that you need "one copy" of the above sequence for each variable; tensoring these all together over $k$ gives you the usual Koszul resolution of $k$ over $k[x_1,...,x_n]$.
It is not hard to pass now to the more general context of elements $a_1,\ldots,a_n$ in a ring $A$, and to imagine the the Koszul complex of $a_1,\ldots,a_n$ will related to the module $A/(a_1,\ldots,a_n)$.  
