Differentiable manifolds can be described in terms of local charts to open subsets of $\mathbb{R}^n$ and transition functions that are diffeomorphisms. Trying to put $\mathbb{A}^n$ (over an algebraically closed field, say) in place of $\mathbb{R}^n$ and isomorphisms of open subvarieties of $\mathbb{A}^n$ in place of local charts of course will not reproduce all the algebraic varieties, even the nonsingular ones, as they have a rich local structure (the stalk of the structure sheaf at a point fully determines its birational equivalence class etc.).

My question is

What about the varieties that

canbe described in this way? At least for the complete ones, is there a characterization or a classification?

Projective spaces and grassmannians belong to this class. Clearly they must be smooth, and birational to $\mathbb{P}^n$ ...