Is there any good definition of many body localization?
Let's start with single-body (Anderson) localization. There, in a non-interacting system, a particle (e.g. an electron) becomes localized due to constructive interference with itself. This interference is induced by the presence of disorder.
Turning interactions on brings us to the realm of many-body physics, as single-particle states are not enough to describe the system anymore. Speaking simply and imprecisely, if behaviour similar to Anderson localization is observed in this many-body system, we say that this system has many-body-localized states or is in the many-body-localized regime.
On a more technical level, I don't think there is a consensus on the strict definition of what a many-body-localized regime is.
One reasonable definition is that many-body localization is present in a many-body system for which the Eigenstate thermalization hypothesis does not hold. Alternatively, a fully many-body localized system is a system where all the many-body eigenstates of the Hamiltonian are localized (see arxiv:1408.4297). There possibly are more definitions, but the differences between them are, in my opinion, small.
It is the property of one state or it is the property of a Hamiltonian?
Since I am not aware of a system where only one single many-body-localized state exists, I would say it is a property of the Hamiltonian of the system.
Why does disorder play an important role in many body localization?
I don't think this is fully understood at the moment.
Starting from Anderson localization, it seems that disorder is important to localize single particles. This property of localization survives perturbative effects of interactions (see e.g. the introduction of this thesis). However, there seem to be systems with no disorder (translationally invariant), where many-body localization is still present! See here for an example.
What is the relation between Anderson localization and many body localization?
Besides the comments already given above, the most important difference between the two is that in a many-body localized situation entanglement spreads in time in contrast to constant entanglement in an Anderson-localized dynamics (also see the table in this review article).