This is another of the early Policies which accelerate military units' production time. It is usually much more useful than Maritime Industries (since making cavalry is always useful); however, note that you will only have real use for it if you truly intend (and need) to build cavalry units, and most of these require Horses Horses. This makes Maneuver most valuable to Scythia, who unlock both Horsemen and Saka Horse Archers with Horseback Riding and receive two of these units for each one they train; and to India and Alexander, both of whose unique units replace the Horseman and have higher Production costs. Egypt will also benefit from Maneuver in Rise and Fall and Gathering Storm, in which their Maryannu Chariot Archer is a ranged cavalry unit rather than a ranged unit. (Be aware, however, that if you adopt both Maneuver and Agoge when playing as Egypt, your Maryannu Chariot Archers will receive benefits from only one of these. The same holds true for Saka Horse Archers when playing as Scythia in Rise and Fall or Gathering Storm.) Otherwise, the only units that benefit from Maneuver are the Heavy Chariot and Sumeria's War-Cart, neither of which have very high Production costs. In this case, unless you're planning a cavalry rush, you'll be better off using your Military Policy slot for something else.
The usual advice when it comes to production-oriented Policies is also applicable here: try to plan in advance when you will produce the units, then make a concentrated effort so that you can later remove the Policy and replace it with a more generally useful one, such as Conscription.
Once warfare moved past the stage of small masses of men just having at it in a club-thumping melee, maneuvers gave the more nimble warriors a distinct advantage in battle. The concept of using rapid movements to keep the enemy at a disadvantage is almost as old as the idea of war itself. Tactical maneuvers are intended to be used on the battlefield, while “maneuver warfare” covers all that leads up to forcing an enemy to fight at a place or time they’d rather not. Both types came to prominence with the advent of chariots and cavalry … and their importance hasn’t diminished since.