Let $\mathbb{A}^n_k$ be the Affine $n$space over an algebraically closed field $k$. Let $X$ be a variety over $k$. What would be the right definition of an "Affine bundle" i.e bundle of fiber type $\mathbb{A}^n_k$ over $X$ (I mean local triviality in zarisky topology,or etale .. )?. When can one get a vector bundle from an "Affine bundle" , more precisely (I think !) if I assume the structure group of the affine bundle to be $Aut_{Var_k}(\mathbb{A}^n_k)$, when can one get a reduction of the structure group to $GL_n(k)$?

I think the term "affine bundle" is used for at least two things: (1) A map $p:Y\to X$ such that for some open cover (in your choice of topology) there are isomorphisms $p^{1}(U)={\Bbb A}^n \times U$  just like you said. (2) A torsor for a vector bundle, i.e., like (1) but with the added condition that the transition functions are affinelinear. In the situation of (2), it's a vector bundle exactly when there's a section (like any torsor). A simple nonvectorbundleexample is the complement of the diagonal in ${\Bbb P}^1 \times {\Bbb P}^1$, projecting onto one of the factors. For (1), I don't know any general (nontrivial) criterion for such a thing to be a vector bundle. (Maybe because the group $Aut({\Bbb A}^n)$ is so complicated...) A simple nonexample is the 2ndorder jet scheme $\mathrm{Hom}(\mathrm{Spec}(k[t]/(t^3)),{\Bbb P}^1) \to {\Bbb P}^1$. The fibers are ${\Bbb A}^2$, and there's a section, but it's not linear. I suppose one test is whether the sheaf of $O_X$algebras $p_*O_Y$ admits a grading generated in degree one. (This fails for the jet schemes, though there is a natural grading by scaling $t$.) 


Dear rvk, as you and Dave said, an affine bundle is a morphism $p:Y\to X$ such that for some open cover $(U_i)$ of $X$ there are isomorphisms $p^{1}(U_i) \simeq {\Bbb A}^n \times U_i$. However the question arises: a cover for what topology? $\:$ In the cases you evoke, étale and Zariski, I am happy to report that it doesn't matter in the case of affinelinear transition functions. In other words an affine bundle locally trivial in the étale topology is already locally trivial in the Zariski topology. The reason is that the affine group $Aff_n (k)$ is "special" in the terminology introduced by Serre here. Serre first proves that $GL_n(k)$ is special (Théorème 2) and then that an extension of a special group by a special group is special (Lemme 6). The specialness of $Aff_n (k)$ (which he doesn't mention) then follows easily from (Proposition 14). Remark 1: It is difficult to overestimate the importance of Serre's article. It might be considered the birth certificate of étale topology under the guise of "fibrés localement isotriviaux". Remark 2 : Groups are rarely special ( Serre knows what words mean!) . For example $PGL_N(k)$ is not special and so bundles with typical fiber $\mathbb P^n$ which are locally trivial in the étale topology needn't be locally trivial in the Zariski topology. This is the subject of Brauer groups of schemes, as envisioned by Grothendieck. 

