Grain defines the direction of the tubular connective tissues in meat. "To cut across the grain" means to slice the connective tissue at right angles. Obviously, the thinner the slices, the shorter the resulting connective tissue segments. The shorter the segments, the less chewy the meat, and therefore the more tender it will seem to the eater.
If you have trouble ascertaining the grain of the meat, try to remember this guideline: Typically, the grain flows in a direction from the middle of the carcass to the animal's extremities (limbs, head, and tail).
** Asian Cooking