Maybe it is helpful to look at these lectures by Yekutieli: arXiv:0801.3233. There he discusses twisted deformations of algebraic varieties $X$ suggested by Kontsevich, i.e. deformations of the category of coherent sheaves on $X$. These deformations are classified by the second Hochschild cohomology of $X$, which by the Hochschild-Kostant-Rosenberg theorem is $H^0(\wedge^2T)\oplus H^1(T)\oplus H^2({\mathcal O})$, where ${\mathcal O}$ is the structure sheaf and $T$ is the tangent sheaf of $X$.
Here, roughly speaking, the first summand corresponds to noncommutative deformations of the structure sheaf (global Poisson bivectors), the second summand corresponds to commutative deformations of this sheaf (i.e., formal deformations of X as a variety), and the third term corresponds to deformations of the category of coherent sheaves which do not arise from deformations of the structure sheaf as a sheaf of algebras (i.e., the algebra deformations exist only on local charts but do not glue into a sheaf; only their categories of modules glue into a sheaf of categories, called a gerbe).
Another helpful reference on this may be van den Bergh's paper arXiv:math/0603200.
