The higher the 'soak' number, the more damage is mitigated (reduced) by the armor. Cutting weapons are not good against armor. Those stats show that 80% of weapon damage is soaked by armor, or I presume slashing damage is soaked by 80% of your armor rating.
An example (40 armor rating):
Cut (vs. unarmored) = 40 damage
Cut (vs. 40 armor) = 8 damage
80% of 40 is 32, subtract this from the damage done (this damage is absorbed by the armor).
Another example (more uneven, 20 armor):
Cut (vs. unarmored) = 40 damage
Cut (vs. 20 armor) = 24 damage
80% of 20 is 16, leaving 24 damage.
If the armor soak is greater than the damage done, you take no damage. If you use the 50% soak from blunt weapons the first scenario takes 20 damage, the second 30. There are a whole lot more factors accounted for in the game however, the damage done is affected by movement speed, the armor rating of the area you hit the enemy, special abilities on the weapon, whether they're defending etc. Beyond that headshots with missile weapons and couched lance damage have their own multipliers.
All the data above is theory from what I understand of damage soak in other games and experience in M&B.
In terms of the different damage types, you can see them on the weapons themselves. Swords will say 23c (the C standing for cutting damage), bows say 14p (piercing) and maces 22b (blunt). There are some weapons that do more than one damage type, depending on how the weapon is used. In general these are the categories.
Cut = swords, axes, glaives
Pierce = bows, xbows, bladed lances, spears, morningstars
Blunt = non-bladed lances, maces, clubs, hammers, staves, swinging long weapons like spears instead of stabbing