Let $R$ be a ring and $x \in R$. The Koszul complex $K_\bullet(x)$ is then $0 \rightarrow R \stackrel{x}{\rightarrow} R \rightarrow 0$. Given $x_1,\dots,x_n \in R$ the Koszul complex $K_\bullet(x_1,\dots,x_n)$ is defined to be $K_\bullet(x_1) \otimes \cdots \otimes K_\bullet(x_n)$, where $\otimes$ is the tensor product of complexes. Then the Koszul homology of an $R$-module $M$ is the homology of the complex $K_\bullet(x_1,\dots,x_n) \otimes M$.
In his notes, Craig Huneke is defining a different Koszul complex $K^\bullet(x)$ by $0 \rightarrow R \rightarrow R_x \rightarrow 0$, where the middle arrow is the canonical map of localization. As above, he defines $K^\bullet(x_1,\dots,x_n) = K^\bullet(x_1) \otimes \cdots \otimes K^\bullet(x_n)$ and the Koszul cohomology of an $R$-module $M$ is the cohomology of the complex $K^\bullet(x_1,\dots,x_n) \otimes M$. In these notes it is then shown that the local cohomology $H^i_I(M)$ of $M$ with respect to an ideal $I=(x_1,\dots,x_n)$ is in fact the Koszul cohomology of $M$.
Question: I have seen before the Koszul homology in Matsumura and Bruns and Herzog, but Koszul cohomology is new to me. The dual denominations Koszul homology-cohomology suggest that the two Koszul complexes $K_\bullet(x_1,\dots,x_n)$ and
$K^\bullet(x_1,\dots,x_n)$ described above are in some sense dual. Is this the case and in what sense?
As Youngsu says, what you have written as Koszul cohomology is sometimes called Cech cohomology instead (geometrically, the terms of degree $> 0$ are exactly those of the Cech complex for the open sets $D(x_i)$ - I will not comment on Huneke's use of the term though).
Here is a way to relate Cech cohomology with Koszul homology: given the Koszul complex $K^\bullet(x) : 0 \to R \xrightarrow{x} R \to 0$, one can form a direct limit of complexes
$$\require{AMScd} \begin{CD} 0 @. 0 @. 0\\ @VVV @VVV @VVV\\ R @>=>> R @>=>> R @>=>> ...\\ @VVxV @VVx^2V @VVx^3V\\ R @>x>> R @>x>> R @>x>> ...\\ @VVV @VVV @VVV\\ 0 @. 0 @. 0 \end{CD} \qquad \implies \qquad \begin{CD} 0 \\ @VVV \\ R \\ @VVV \\ R_x \\ @VVV \\ 0 \end{CD}$$
where the vertical columns are $K^\bullet(x^j)$, and the limit is the Cech complex, which I will call $C^\bullet(x) = \varinjlim K^\bullet(x^j)$. Tensoring such sequences together gives (since $\varinjlim$ and $\otimes$ commute) $C^\bullet(x_1, \ldots, x_n; M) \cong \varinjlim K^\bullet(x_1^j, \ldots, x_n^j; M)$. One can show that the cohomology of this complex computes local cohomology, i.e. if $I := (x_1, \ldots, x_n)$, then $H^i_I(M) \cong H^i(C^\bullet(x_1, \ldots, x_n; M))$, and since $\varinjlim$ is exact, this is $\varinjlim H^i(K^\bullet(x_1^j, \ldots, x_n^j; M))$ (in the case that $x_1, \ldots, x_n$ is a regular sequence, one can see this directly by noting that the Koszul complex on $(x_1^j, \ldots, x_n^j)$ is a free resolution of $R/(x_1^j, \ldots, x_n^j)$, thus can be used to compute $\text{Ext}$, and local cohomology is a limit of $\text{Ext}$'s.
For more information on this (such as the relation to sheaf cohomology), see e.g. Appendix 4 in Eisenbud.