"Minkowski" is what corresponds to "flat" in GR spacetime it is the geometry of Minkowski's original spacetime of special relativity; the curvature of a Minkowski spacetime is zero. Asymptotically Minkowski means the curvature of spacetime falls off toward zero as you move far enough in any direction.

A hyperbolic space would be an example. It has a region of maximum curvature and as you move farther and farther from that region the curvature gradually tends toward zero.

Things like energy, mass and momentum etc. are quite tricky to define in GR.

Usually we get conserved quantities from a finite number of continuous symmetries (i.e. Noether's theorem). But in GR, we have an infinite number of symmetries -- we can transform to any co-ordinate system we choose and the physical result is still the same. I believe this is where the problem arises from.

Even in classical physics momentum and energy can only be defined with respect to a particular frame of reference. (If an observer on a moving train and one on the ground measure the kinetic energy and momentum of the train, the first observer will find both to be 0, while the second will find answers depending on the train's speed.)

This problem is much compounded in GR because there are no universal frames of reference (exceping in rare special cases). You can still define a "local inertial frame" around a given point; and, given that definition, you can speak about the momentum and energy of an object at the point in that frame. And, the procedure for transforming to a different frame is also quite straightforward. However, once the particle moves to a different point, momentum and energy must be redefined for a local inertial frame about the new point.

You need either an asymptotically flat (or as the wiki says, an asymptotically Minkowskian) space-time or space-translation symmetry before you can define momentum.

If you had a system of masses, alone, in a totally empty universe without a cosmological constant, space-time would automatically be asymptotically flat. And then you could define the momentum of the system.

However, if you have a system with space-translation symmetry, you can also define a conserved momentum for the system. This is handy, because the FRW cosmology that is felt to represent our universe is not asymtotically flat, but it does have space translation symmetry. Therfore we can define a conserved momentum.

As far as "why" goes, look up Noether's theorem (in the wiki & elsewhere).

You might also look at the wiki article http://en.wikipedia.org/wiki/Mass_in_General_Relativity which also talks about some of these issues (it's title is about mass, but it also talks about momentum and energy) and it has some references which talk about Noether's theorem.

FYI: I should probably disclose that I'm the primary author of the above article.