Flaming weapons are either melee weapons of missile weapons that cause fire damage to the target. This damage is in addition to other damage caused by the weapon, if any. When a model is hot by a flaming attack, the Flaming Weapon entry will tell you the dice roll to make in order to see if the target catches fire. For example, Flaming Weapon(4+) indicates that the target will catch fire on a 1d6 roll of a 4+ when hit. If a target catches fore, they suffer 1d6 Wounds with no modifiers for Toughness or Armour at the start of each subsequent turn. Mummies, Tree-kin like Dryads and Treemen and other dry, flammable targets suffer 2d6 damage each turn instead.
If the damage dice comes up as a natural 1 or 2, then the flames go out and no more fire damage is taken in subsequent turns.