Physical intuition for deformation quantization of Poisson manifolds First of all, I know almost nothing about physics. I was reading Kontsevich´s paper on Deformation quantization of Poisson manifolds, however I could not figure out what´s the intuition for such operation. 
Why there is the word "quantization" and the Planck constant in the star product? Where's the physics in such formal deformation of the algebra $\mathscr{A} = \mathscr{C}^{\infty}(M)$ ?
In fact, the unique thing that I can understand is the "deformation" part, since (if I'm not wrong) it looks like a deformation of $\mathscr{A}$ along the 2-cocycle $\{ \cdot, \cdot\}$ of the Hochschild cochain complex.
Another thing, that I never got: why is the deformation done only in the global sections and not in the whole sheaf $\mathscr{C}^{\infty}$?
 A: Quantization usually means the association of a Hilbert space to the classical phase space (in our case a Poisson manifold). However, in deformation quantization, this task is achieved indirectly, first through the construction of an associative $C^*$ algebra, in this case the deformed algebra of functions equipped with a star product which serves as the associative product of the $C^*$ algebra. This algebra is dependent on a formal parameter $\hbar$. Once an associative $C^*$ algebra is constructed, a Hilbert space representation can be constructed in principle by $C^*$ algebraic techniques such as the GNS construction. Please see for example the following article  by Stefan Waldmann.
The motivation of deformation quantization is that in many physical models, the Taylor series with respect to the Planck's constant $\hbar$ gives a viable deformation quantization. The prototype of an explicitly known star product whose Taylor series in $\hbar$ defines a deformation quantization is the Moyal product on $\mathbb{R}^{2n}$. Also, there is the Gutt star product on the duals of Lie algebras, please see the following article by  Monvel. There are also, the Wick and Anti-Wick star products and their
generalizations in Berezin quantization of Kähler manifolds. Please, see for example this article  by Bordemann, Brischle,  Emmrich, Waldmann. Another known construction of
geometric origin is the Fedosov star product, please see the following
Philip Tillman thesis by .
The Hoschild closure of the deformation chains is needed to ensure the associatively of the star product, please see the following article by: McCurdy and Zumino (although treating a special case, but the relation between closure and associatively is clarified).
In Kontsevich construction, the sections need to be global, because the construction was performed locally for $\mathbb{R}^d$, and there is a need to globalize to a general Poisson manifold.
