When you get a weapon, it will have a range of damage. The maximum damage on a bow or axe is 28. Swords, staffs and wands max out at 22. Hammers max out at 35.

Say your new sword has a range of 11-18 and requires 5 swordsmanship. That means you must have a minimum of 5 swordsmanship to get 11 damage out of the weapon. The damage goes up for every point of swordsmanship you have above 5 up to 18 damage at 12 points of swordsmanship. At 10 points of swordsmanship, you will get about 91% of the high damage number, or 16 damage. At 16 points of swordsmanship you will get about 115% of the high number, or 20 damage.

Weapons can generally take two upgrades. Wands can't have any upgrades. You will see weapons like Fiery Long Sword of Swordsmanship. The unaugmented sword would be simple a long sword. The Fiery comes from an upgrade (usually a sword hilt) that adds fire damage. The Swordsmanship comes from an upgrade (usually a pommel) that adds an amount to your swordsmanship, but there is usually a percentage chance on these additions. Thus you might see "swordsmanship +1 (11% chance)" On average, this would give you the added damage of one more level of swordsmanship every 1 in 10 hits.