How should I think about B-fields? So, physicists like to attach a mysterious extra cohomology class in H^2(X;C^*) to a Kahler (or hyperkahler) manifold called a "B-field."  The only concrete thing I've seen this B-field do is change the Fukaya category/A-branes: when you have a B-field, you shouldn't take flat vector bundles on a Lagrangian subvariety, but rather ones whose curvature is the B-field.  How should I think about this gadget?
 A: Let me add a few words of explanation to Aaron's comment.  Perturbative string theory is (at least at the level of caricature) concerned with describing small corrections to classical gravitational physics on the spacetime X.  So, to do perturbative string theory on X, you need to choose a "background" metric on X.  You might need to choose other fields as well, but we can assume for now that those are all set to zero.  
Having chosen a metric, you can talk about strings moving in X.  In the limit where the string length goes to zero, a single string will look like a particle.  What sort of particle it looks like will depend on how it's vibrating inside X.  In particular, a closed string has a set of vibrational states which a) appear massless in this limit, and b) fill out a representation R of the Lorentz group.   Specifically, R is the representation induced from the tensor square V (x) V, where V is the standard representation of the little group that fixes some light-like vector.   You can decompose V into a sum of traceless symmetric square, trace, and antisymmetric traceless square.  The states in the first summand are states of the graviton, representing tiny quantum excitations of the metric in X.  The states in the last summand, the antisymmetric representation, are tiny excitations of the B-field, which we set equal to zero.  (The states in the trace representation are quanta of the "dilaton" field.)
So, we didn't give the B-field any respect when we started, but it turns out to part of the definition of a string background.  And once you know about the B-field, it's easy to include it in the action for the sigma model to X:  Add to your action the term i<[S],f*B>, where [S] is the fundamental class of the Riemann surface, and f: S -> X is the function embedding your string's worldsheet into X. Edit:  Forgot a factor of i=root(-1), which is necessary to make the action real.  
And I forgot to mention that Aaron's H is dB. 
A: Let me try to add a different point of view on B-fields and mirror symmetry. Ideally in mirror symmetry, given a Calabi-Yau manifold X, you would like to "construct" its mirror X', where the symplectic form on X should give you the complex structure on X'. As already mentioned, classes of symplectic forms have moduli of real dimension $h^{1,1}(X)$ and complex structures on X' have moduli of complex dimension $h^{2,1}(X') = h^{1,1}(X)$. So the kahler class is not enough to determine all complex structures on X'. In the context of the Strominger-Yau-Zaslow conjecture there is a nice interpretation of the B-field. Suppose X = $T^*B / \Lambda$, where B is a smooth manifold and $\Lambda$ is locally the span over the integers of 1-forms $dy_1$, ..., $dy_n$ (here $y_1$, ..., $y_n$ are coordinates which 
change with affine transformations from one chart to the other). Then $X$ has a standard symplectic form. We can consider $X'= TB / \Lambda'$, where  $\Lambda'$ is the dual lattice. 
Then X' has a natural complex structure defined as follows. In standard coordinates on TB, given by $(y,x)$ --> $x \partial_y$, the complex coordinates on X' are $z_k = e^{2\pi i(x_k + i y_k)}$, which are well defined due to the nature of the coordinates x and y. But the above complex coordinates can be twisted locally (on a coordinate patch) by  $z_k (b) =  e^{2\pi i(x_k + b_k  + i y_k)}$, where $b = (b_1, \ldots, b_n)$ is some local data. But since on overlaps $U_i \cap U_j$ the coordinates have to match, we must have $b(i) - b(j) \in \Lambda$. It turns out that by putting $b_{ij} = b(i) - b(j)$ on overlaps, we get a cohomology class in $H^{1}(B, \Lambda)$, this is the B-field. The cohomology group $H^{1}(B, \Lambda)$ shoud coincide (in some cases at least) with $H^2(X, R/Z)$, which is what Kevin Lin mentioned. The elliptic curve case (mentioned by Kevin) can be seen from this point of view. 
This point of view is also called "mirror symmetry without corrections" and it only approximates what happens in compact Calabi-Yaus. I have learned this in papers by Mark Gross (such as "Special lagrangian fibrations II: geometry") or the book "Calabi-Yau manifolds and related geometries" by Gross, Huybrechts and Joyce. 
I would be interested to know how this interpretation connects to the other ones which have been described. 
A: We like to do more than that, actually. The B-field is an element in the differential cohomology class $\check{H}^3(M)$, or, more geometrically, a connection on an abelian gerbe. Thus, there is a class $[H] \in H^3(M,Z)$ characterizing the gerbe. In the B-model, this twists the derived category. The connection is the part that changes the A-model, and when $[H] = 0$, you exactly get that the differential cohomology group is $H^2(X,U(1))$. In the geometric language, it's a flat connection on a trivial gerbe.
A: Just to supplement Aaron's and A.J.'s comments: nLab:Kalb-Ramond field
A: In general, you should think about B-field in quantum field theory as providing some noncommutativity.
