There's a kind of analogy between the way manifolds work and the way bundles work. Let me try to give some examples of the analogy (although there may be better ones). I'll stick to smooth manifolds and vector bundles for definiteness.

(the main idea) A manifold is $M$ a space with an open cover $\{U_i\}$ by copies of the model space $\mathbb{R}^n$, with transition functions coming from the diffeomorphism group. A vector bundle $V \to X$ is a map where the base $X$ has an open cover $\{U_i\}$ on which the map pulls back to copies of the standard map, which is a projection from $\mathbb{R}^n \times U_i \to U_i$, with transition functions required to be fiberwise in the general linear group.

An orientation on a (compact) manifold $M$ is a choice of a fundamental class, i.e. $[M] \in H_n(M)$ which restricts to a generator on the local homology group $H_n(M,M-x)$ at each point $x \in M$. An orientation on a vector bundels $V \to X$ is a choice of Thom class, i.e. $v \in H^n(V,V-0)$ (where $0$ is the zero section of $V$) which restricts to a generator on the local cohomology group $H^n(V_x, V_x - 0)$ for each point $x \in X$, where $V_x$ is the fiber at $x$.

Manifolds have a Pontrjagin-Thom construction, while vector bundles have a Thom construction. The idea that you can "collapse at infinity" is something in common. This connection may be me grasping at straws -- anyway, the two constructions are already closely related: the Thom construction of the normal bundle receives the map from the manifold in the Pontrjagin-Thom construction.

(attempt at (1) in a more Cech-like language:) A manifold is some kind of simplicial object of model spaces. A vector bundle is some kind of simplicial object of maps to a structure group (satisfying a cocycle condition).

(attempt at (1) in a more sheaf-like language:) I think a manifold $M$ is some kind of locally free sheaf on the site of Euclidean spaces and $C^\infty$-functions. The sheaf of sections of a vector bundle $V \to X$ is a locally free module over the sheaf $C^0(X)$ of continuous functions on $X$.

Point (1), in particular, leads me to want to view bundles as a "relative version of manifolds" (although vector bundles in particular happen to have a much smaller structure group). I suppose all of this could be considered in other types of geometry, too -- e.g. the same sort of analogy holds between schemes and certain kinds of sheaves over schemes. But I'd like to be able to state this in some precise way. I'd also like to understand whether the analogy stops there, or whether there's a whole hierarchy of notions of which these are just 0th-order and 1st-order notions.