What is the relation between vector bundles on a manifold and grassmanians? is there a relation between vector bundles on a manifold and plücker embeddings
 A: It is fun to give an answer when there is no question :-)) LOL...
Anyway, pick your vector bundle $V$ on a  manifold $X$ and consider its top exterior power $\Lambda^m V$. Now you have Plucker map from $X$ to the projective space $P(\Gamma (X,\Lambda^m V)^\ast)$ of the dual space of the global sections. It is given by mapping of $p\in X$ to the restriction map of the global sections $\Gamma (X,\Lambda^m V)\rightarrow \Lambda^m V_p$. Notice that $\Lambda^m V_p$ is a one-dimensional vector space without a natural basis - different choices of basis give different functionals, so the map naturally goes to projective space.
You'd better have your global sections finite-dimensional, which you get by adding holomorphicity + compactness. If you want embedding, you have to play with ampleness, etc.
A: If $V$ is any finite dimensional vector space over a field $k$ and $r$ is any integer between 0 and $\dim(V)$, you can see the Grasmannian $Grass(V, r)$ as the functor
$(Sch/k) \to (Sets)$ which associates to any k-scheme of finite type $S$ the set of all subsheaves $K \subset \mathcal{O}_S \otimes_k V$ such that the quotient $F:=\mathcal{O}_S \otimes_k V / K$ is locally free of rank $r$. In this way the Plucker embedding is given by
$Grass(V, r) \to  \mathbb{P}(\Lambda^r V)$
$[\mathcal{O}_S \otimes_k V  \to F] \to [\mathcal{O}_S \otimes_k \Lambda^rV \to \det(F)]$.  
This description is slightly less intuitive than the "classical" Grasmannian (but of course it is just its relative version); on the other hand it can be naturally generalized to the case where $V$ is replaced by a coherent sheaf $\mathcal{V}$ on $S$, in particular by a vector bundle. See the book of Huybrechts and Lehn "The geometry of moduli spaces of sheaves", p. 41 for further details.
A: I note that it is possible to move over from the manifold point of view, to use the more refined approach of algebraic geometry. This is because the Plucker embedding is actually "algebraic". 
In that setting, Grothendieck's Quot scheme would classify quotients(or, looking at the kernel, subbundles) of a free vector bundle. Actually, we use locally free sheaves instead of vector bundles. Indeed, this Quot scheme is stratified for each polynomial, classifying the quotients with a certain "Hilbert polynomial". 
And then the Grassmannian is just a particular component of the Hilbert scheme, corresponding to just one particular Hilbert polynomial. The construction of this particular case is precisely via the Plucker embedding, realizing the Grassmannian as a projective variety. On the other hand, in the construction of Quot scheme I have read, it is constructed as some subscheme of the Grassmannian. In this sense, the Plucker embedding is very intertwined with the study of vector bundles.
(Here one might note that this answer is somewhat similar to Francesco Polizzi's).
Added note(prompted by BCnrd's comments below): If you are not into algebraic geometry, please forget completely about this post. If you ever get into it, then you might come back and read it again.
