There are two types of bordism groups: bordism groups, also called cobordism groups or cobordism rings, and there are singular bordism groups. The bordism groups give a framework for getting a grip on the question, "When is a compact boundaryless manifold the boundary of another manifold?" The answer is, precisely when all its Stiefel-Whitney numbers are zero. Singular bordism groups give insight into Steenrod's realization problem: "When can homology classes be realized as the image of fundamental classes of manifolds?" That answer is known, too.
The machinery of the bordism group winds up being important for homotopy theory as well.