Twisted sheaf $\mathcal{F}(n)$. Let $\mathcal{F}$ be a sheaf on a scheme $X$ and $O_X(k)$ as usual. We define $\mathcal{F}(n) = \mathcal{F} \otimes_{O_X} O_X(n)$, I don't undertand this definition. What is this tensor product? Then, if we can try to find examples, if $n=1$ and $\mathcal{F}=\mathcal{O}_{\mathbb{A}^1_k}$, who is $\mathcal{O}_{\mathbb{A}^1_k}(1)$?, and  $\mathcal{O}_{\mathbb{A}^1_k}(n)$?

I am sorry, but I think $O_X(n)$ can be defined when the underlying scheme is projective.
So I think the proper example can be $\mathcal{O}_{\mathbb{P}^1_k}(1)
$,and this is just tautological line bundle which is made from the dgree one parts of the oringinal graded ring.  
