Given a variety V and a locally free (coherent) sheaf $\mathcal{F}$ of rank 1 (equivalently a line bundle $L$), I can do a Cech cohomology on it. Then $H^0(V; \mathcal{F})$ are just global sections. Is there a similarly understandable meaning to elements of $H^1(V; \mathcal{F})$?
Thanks!
$H^1(V;\mathcal{F})$ 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 $H^1(V;\mathcal{O}_V^*)$.
$H^1$ is the first derived functor of the functor $H^0$ of global sections.
In the Cech cohomology construction, note that we look whether the local sections glue together to form global sections. On an affine space, this is indeed true. But, not so in general. We would like to address the obstruction using algebraic means. When things glue well, we have an exact sequence. In homological algebra, the question of exact sequences breaking down under a functor is addressed by the machinery of "derived functor".
So $H^1$ and the higher cohomology groups are in a sense the obstruction to local sections patching up to form global sections. Since $H^0$ behaves well on affine spaces, it in a sense measures the failure of affineness also. It captures geometry by seeing how the affine pieces glue together to form a projective variety, for instance. The dimensions in which geometry is interesting can be seen by at which dimension the derived functors are nontrivial.
This is my personal point of view to see how geometry is involved, based on derived functors.
