The "modernization" of Egypt's army in the New Kingdom began after the Egyptians fought and conquered the Hyksos who had occupied parts of Lower Egypt, sometime shortly after 1600 bce.
The Egyptian Army was organized in ways that seem familiar to us in the 21st century: the commander-in-chief of the army, for example, was usually the current pharaoh (who had a personal bodyguard loyal only to the pharaoh), and he had a staff of what we would now call "general" officers who commanded army divisions of approximately five thousand very well-trained, professional troops. As with modern armies, divisions were the largest tactical unit and were subdivided into much small units of perhaps 300 soldiers.
The principal unit of the New Kingdom army was, like that of modern armies, the infantry, which was generally composed of archers, spear-carriers, axe-men, and slingers (literally, those who used the slingshot). Unlike armies in later periods, Egyptian New Kingdom armies did not have cavalry as such, but they did have charioteers, whose primary jobs were to provide intelligence of an enemy's strength and movements, as well as to protect the infantry from attacks by the enemy's charioteer force.
Several modifications to Egyptian chariots--moving the rear axle further back, for example--made the chariot force in New Kingdom armies universally better than other charioteer troops of the time because the Egyptian chariot was stronger, more maneuvarable, and, more important, much more stable, which contributed to the chariot force's increased lethality over its enemies. From a tactical and strategic standpoint, then, a New Kingdom army had a significant advantage with the most important weapons system of the time.
Auxiliary troops--from conquered states such as Libya and Nubia--were also an important and integral part of the organization of the New Kingdom army, but because such troops essentially fought as mercenaries, they were not given the most critical assignments in battles and were, more often than not, used as "cannon fodder" to keep the enemy occupied while more skilled troops attacked at critical points in the enemy's lines.
In addition to technologically-advanced chariots, the New Kingdom armies benefited from better armor than their enemies used, as well as the compound bow (sometime after 1600 bce), which increased the bowmen's rate of fire (because the bow was smaller and more maneuverable), accuracy and distance.
From a purely military perspective, the success of New Kingdon armies was more of a result of technological advances in weaponry, armor, and tactics than in simple organizational matters. Most armies fighting against Eqyptian New Kingdom armies were similarly organized but did not avail themselves of advances in weapons technology to the extent that the Egyptians did.