Rene Thom answered this in section II of "Quelques propriétés globales des variétés différentiables." Every class x $x$ in Hr(X; ZZ) $H_r(X; \mathbb Z)$ has some integral multiple nx $nx$ which is the fundamental class of a submanifold, so the homology is at least rationally generated by these fundamental classes.
Section II.11 works out some specific cases: for example, every homology class of a manifold of dimension at most 8 is realizable this way, but this is not true for higher dimensional manifolds and the answer in general has to do with Steenrod operations.
