This is exactly like rock/paper/scissors. The Weapon Triangle influences the battle by giving one weapon type an advantage over another. For example, if Character 1 uses a sword and Character 2 uses a lance, I will have the disadvantage, as shown below by "Lance beats sword". However, if Character 1 fights Character 3, he will have an advantage because character 3 is fighting with an iron axe. An advantage does slightly more damage and has a better chance of hitting the opponent; during a disadvantage there is a less chance of hitting and the attack does slightly less damage. This can make a significant difference during the course of a battle.

Sword beats axe
Axe beats lance
Lance beats sword

Where are the bows you ask? Well they are left out of the triangle because they can only be fired at a distance and usually can't be countered against.