There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$,


* a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$
* or an almost complex structure is a section $J$ of $\textrm{End}(TM)$ which is everywhere an anti-involution (*i.e.* $J\_x^2 = - \mathrm{Id}\_{T\_x M} $)
* or an orientation is a non-vanishing section of $\Lambda^m TM$


is 


a spin structure, a section of quadratic forms $Q$ on $TM$ (of type $(s,t)$) and a vector bundle $S$ so that $\textrm{End}(S) \simeq \textrm{C}\ell(TM,Q)$?


...or something of the like? any references where it may be stated in this fashion?



what is a spinor structure?
Chapter 9 of *Elements of Noncommutative Geometry,* by Gracia-Bondia, Varilly, and Figueroa, has this perspective on spin$^c$ and spin structures. 


The way to think about this algebraically is that the module of (continuous, say) sections of a spinor bundle over a (compact, Riemannian) manifold $M$ is Morita equivalence bimodule for the algebras $C(M)$ and $Cl(M)$, where $C(M)$ is the algebra of continuous functions and $Cl(M)$ is the algebra of continuous sections of the Clifford bundle (formed using the Riemannian metric). You can replace "continuous" with "smooth" here with no problems.


