A spacetime is asymptotically flat if far enough away from any point it looks like flat Minkowski spacetime.
On asymptotically flat spacetimes a total gravitational mass/energy is well defined, the ADM mass?. See also at positive energy theorem.
Regarding a field configuration of Einstein gravity as a Cartan connection for the inclusion of the Lorentz group into the Poincaré group (see at first-order formulation of gravity), then asymptotic flatness means that the Cartan geometry asymptotivally approaches its model Klein geometry. From this perspective there are various generalizations of asymptotic flatness:
for instance for anti de Sitter gravity it means that spacetime asymptootes to anti de Sitter spacetime. For more on this see at AdS-CFT correspondence.
Wikipedia, Asymptotically flat spacetimes
Jörg Frauendiener, Conformal Infinity (living reviews)