The base manifold in a bundle is analogous to the domain for a set of functions. In fact, a bundle, by definition, comes with a map to the
base manifold, often called 
or projection.
For example, the base manifold to the tangent bundle of a manifold 
is the manifold 
. A vector field is a function
from the manifold to the tangent bundle, with the
restriction that every point gets mapped to a vector at that point. In general, a
bundle has bundle sections,
at least locally, which are maps from the base manifold to the bundle.
