I always encounter two definitions: two-sided and oriented (hypersurface or submanifold). What is the difference of them? Which one is stronger?



Difference of two-sided and oriented
If the submanifold $M$ of a manifold $N$ is co-dimension one, being *two-sided* typically means it has a trivial normal bundle, i.e. $M$ splits the normal bundle into two path components. These are the two sides. 


Technically, being two-sided is unrelated to being oriented, but there are cases where they are related. 


For example, take the central circle in the Moebius band. The circle is orientable, the Moebius band is not, the normal-bundle is not trivial. This is in the one-sided case.


Or take the Moebius band as the 0-section in the 1-dimensional bundle (with orientable total space) that contains it. The Moebius band is not orientable. The total space of the 1-dimensional bundle over the Moebius band is orientable. This is again a 1-sided case. 


If the ambient manifold is orientable (or really, just the total space of the tubular neighbourhood), then the triviality of the normal bundle is equivalent to the orientability of the co-dimension one submanifold. 


