Quality - the direct modifier to how much damage your weapon does.
Condition - another direct modifier, but one that kicks in AFTER quality.
Durability - has no effect on anything but how long a weapon will remain in the world.
So, all other things being equal (and dozens of factors go into the damage rolls, I can't explain them all without some serious mathematics, and guys, cut me some slack, I majored in theatre and secondary English education), if you (with your skill in the weapon type) wielding the weapon could do 100 damage, but the quality and condition of the weapon were both 80%, then the weapon would first be cut to 80 (quality) and then 80% of 80 (the second 80% being the weapon's condition).
Bonus - This takes into account any bonuses on the armor of the person you're hitting. I'll use the simplest possible example. Let's say your opponent is wearing a shirt with a 10% bonus. You have a sword with a 15% bonus. What you then have is a 5% addition to your "to hit" (which is simply your chance to hit the opponent). It doesn't affect how MUCH you hit for, it increases your chances of landing a hit on him at all.
Your specialization in the weapon type determines your "range" - the number between the minimum damage the weapon can do and the maximum. Let's say someone with no spec in sword can hit, with a certain type of sword, for as low as 5 points and as many as 20. If you have points in that type, you could still only hit for a max of 20, but your damage floor would be higher - let's say 10. The more spec points, the higher the floor. So being speced in a weapon type means you will over time do a much higher average damage than someone who is untrained.
The whole system is an attempt to make it as realistic as possible.