In a physics article I read recently, the author introduces the notion of supersymmetry by saying basically that the system described by the Hamiltonian $H$ is supersymmetric if $H$ can be decomposed as $H=QQ^\dagger + Q^\dagger Q$ with $Q^2=0$. This condition then leads to a nice splitting of the underlying Hilbert space, which seems to help a lot when working with $H$. Even from the point of view of simple linear algebra, this property is obviously interesting.
Then, I heard that the symmetry is said to be broken or unbroken depending on whether or not $H$ has any ground states, or in other words if the null space of $H$ is non-trivial. I can understand this as a definition, but why would this be interesting? In particular, what consequences are there to supersymmetry being (un)broken (preferably at the low level)? Which systems are "better"?