I would like to understand is the difference between the line bundle $\mathcal{O}_X$ and the fiber bundle $\mathcal{O}_X(n)$ which can be written as $\bigotimes_n \mathcal{O}_X(1)$ on $\mathbb{P}^1$. $=\mathbb{P}^1$ for all purposes.
So to start with $\mathcal{O}_X$ which is the same as $\mathcal{O}_X(0)$ and this denotes the trivial line bundle on $\mathbb{P}^1$ or the structure sheaf of regular functions. In what way is $\mathcal{O}_X(1)$ different then and in what sense it is dual to $\mathcal{O}_X(-1)$?
The other question is why can any, say rank 2 vector bundle $E$ be written as the tensor sum $E = \mathcal{O}(n) \oplus \mathcal{O}(n)$?
I would like for an us much as possible intuitive answer since I struggle a lot with these concepts.