A presentation of a affine complex variety consists of finitely many polynomials $f_1,...,f_m$ in $\mathbb{C}[x_1,...,x_n]$. A presentation of a projective complex variety consists of finitely many homogeneous polynomials $f_1,...,f_m$ in $\mathbb{C}[x_1,...,x_{n}]$.
Is there something similar for smooth manifolds and power series? Specifically, I would like to figure out how to think about "presentations" for smooth manifolds using power series.
What is a notion of presentation for smooth manifolds and their bundles?