TITLE: Interesting vector bundles other than the tangent and cotangent bundle
QUESTION [5 upvotes]: Given a real/complex topological/smooth manifold $M$, there are two natural vector bundles that we are interested in :

Tangent bundle $\pi: TM\rightarrow M$
Cotangent bundle $\pi^*:(TM)^*\rightarrow M$

I would like to know if there are any natural/interesting vector bundles that we would come across if we go further in Algebraic/Differential Geometry.
I am almost sure that this question is not well posed, but I am not able to make this any better.

REPLY [6 votes]: If $M$ admits a spin structure (i.e. $w_1(M) = 0$ and $w_2(M) = 0$), then one can form the spinor bundle $\mathbb{S} \to M$ which is a complex vector bundle. 
A spin structure on $M$ is a choice of principal $\operatorname{Spin}(n)$-bundle  which is an equivariant lift of the oriented orthonormal frame bundle (a principal $SO(n)$-bundle). The spinor bundle is the complex vector bundle associated to this principal $\operatorname{Spin}(n)$-bundle via a complex representation of $\operatorname{Spin}(n)$ which does not factor through $SO(n)$.
The construction of this bundle is the beginning of spin geometry. A reference for this material is Lawson and Michelsohn's Spin Geometry.