The relationship between weapons doesn't completely determine who will win, but it is a major influence. During battle, you can order your troops to take an aggressive or passive stance, which can sometimes give even a weapon on the "weak" end of the relationship a little extra advantage. Some weapons, like bows, have special traits depending on how they are used in battle. For instance, bows are weak against everything when used on defense, but powerful on offense. The turn's governing card also gives you a special skill to use in battle if you can fill a small meter at the top of the screen. The catch is that this meter also fills or depletes based on how passively or aggressively you approach a battle.
Yggdra Union tosses out the concept of hit points and replaces them with morale. Individual battles are fought on a side-by-side screen similar to Advance Wars. The size of the army on each side of the screen gives a reasonably good idea of each army's health, though it can sometimes be tough to tell who is winning. Troop numbers are refreshed each turn, giving you even more reason to use formations. Even if one of your units is defeated, another can swoop in and destroy what's left of the victor. After the battle, the losing unit takes a hit to their morale number and once it hits zero, they disappear from the battlefield.