A $B$-bordism is a cobordism $W$ equipped with extra “topological structure” with respect to (B,f)-structure, i.e. in the form of a lift of the classifying map $W \to B O$ of its tangent bundle/stable normal bundle? through some fibration $B \to B O$ over the classifying space of the orthogonal group.
Commonly considered are lifts through the Whitehead tower of the orthogonal group), corresponding, in this order, to cobordisms with
fivebrane structure - fivebrane bordism?
etc.