$\newcommand{\O}{\mathcal{O}}$
$\newcommand{\F}{\mathcal{F}}$
The way you get a locally free sheaf of rank $n$ from a $GL(n)$-torsor $P$ is by twisting the trivial rank $n$ bundle $\O^n$ (which has a natural $GL(n)$-action) by the torsor. Explicitly, the locally free sheaf is $\F=\O^n\times^{GL(n)}P$, whose (scheme-theoretic) points are $(v,p)$, where $v$ is a point of the trivial bundle and $p$ is a point of $P$, subject to the relation $(v\cdot g,p)\sim (v,g\cdot p)$. Conversely, given a locally free sheaf $\F$ of rank $n$, the sheaf $Isom(\O^n,\F)$ is a $GL(n)$-torsor, and this procedure is inverse to the $P\mapsto \O^n\times^{GL(n)}P$ procedure above. (Note: I'm identifying spaces over the base $X$ with their sheaves of sections, both for regarding $Isom(\O^n,\F)$ as a torsor and for regarding $\O^n\times^{GL_n}P$ as a locally free sheaf.)
Similarly, if you have a group $G$ and a representation $V$, then you can associate to any $G$-torsor $P$ a locally free sheaf of rank $\dim(V)$, namely $V\times^G P$. But I don't know of a characterization of which locally free sheaves of rank $\dim(V)$ arise in this way.
Operations with the locally free sheaf (like taking top exterior power, or any other operation which is basically defined fiberwise and shown to glue) correspond to doing that operation with the representation $V$, so I think you're right that in the case of $SL(n)$ you get exactly those locally free sheaves whose top exterior power is trivial (since $SL(n)$ has no non-trivial $1$-dimensional representations).
The bundles with no derived global sections (more generally the objects $F$ of the derived category $D^b(coh X)$ such that $Ext^\bullet(O_X,F) = 0$) form the left orthogonal complement to the structure sheaf $O_X$. It is denoted $O_X^\perp$. This is quite an interesting subcategory of the derived category.
For example, if $O_X$ itself has no higher cohomology (i.e. it is exceptional) then there is a semiorthogonal decomposition $D^b(coh X) =< O_X^\perp, O_X >$. Then every object can be split into components with respect to this decomposition and so many questions about $D^b(coh X)$ can be reduced to $O_X^\perp$ which is smaller. Further, if you have an object $E$ in $O_X^\perp$ which has no higher self-exts (like $O(-1)$ on $P^2$), you can continue simplifying your category --- considering a semiorthogonal decomposition $O_X^\perp = < E^\perp, E >$. For example if $X = P^2$ and $E = O(-1)$ then $E^\perp$ is generated by $O(-2)$, so there is a semiorthogonal decomposition $D^b(coh P^2) = < O_X(-2), O_X(-1), O_X >$ also known as a full exceptional collection on $P^2$. It allows a reduction of many problems about $D^b(coh P^2)$ to linear algebra.
Another interesting question is when $O_X$ is spherical (i.e. its cohomology algebra is isomorphic to the cohomology of a topological sphere). This holds for example for K3 surfaces. Then there is a so called spherical twist functor for which $O_X^\perp$ is the fixed subcategory.
Thus, as you see, the importance of the category $O_X^\perp$ depends on the properties of the sheaf $O_X$.
Best Answer
$H^1(V;\mathcal{F})$ is the space of bundles of affine spaces modeled on $\mathcal{F}$. An affine bundle $F$ modeled on $\mathcal{F}$ is a sheaf of sets that $\mathcal{F}$ acts freely on as a sheaf of abelian groups (i.e., there is a map of sheaves $F\times \mathcal{F}\to F$ which satisfies the usual associativity), and on a small enough neighborhood of any point, this action is regular (i.e., the action map on some point gives a bijection). You should think of this as a sheaf where you can take differences of sections and get a section of $\mathcal{F}$.
This matches up with what Anweshi said as follows: given such a thing, you can try to construct an isomorphism to $\mathcal{F}$. This means picking an open cover, and picking a section over each open subset and declaring that to be 0. The Cech 1-cochain you get is the difference between these two sections on any overlap, and if an isomorphism exists, the difference between the actual zero section and the candidate ones you picked is the Cech 0-chain whose boundary your 1-cochain is.
Another way of saying this is that a Cech 1-cycle is exactly the same sort of data as transition functions valued in your sheaf, so if you have anything that your sheaf acts on (again, as an abelian group), then you can use these transition functions to build a new sheaf; a homology between to 1-cycles (i.e. a 0-cycle whose boundary is their difference) is exactly the same thing as an isomorphism between two of these.
I'll note that there's nothing special about line bundles; this works for any sheaf of groups (even nonabelian ones). For example, if you take the sheaf of locally constant functions in a group, you will classify local systems for that group. If you take continuous functions into a group, you will get principal bundles for that group. If you take the sheaf $\mathrm{Aut}(\mathcal{O}_V^{\oplus n})$, you'll get rank $n$ locally free sheaves. A particularly famous instance of this is that line bundles are classified by $H^1(V;\mathcal{O}_V^*)$.