The ability to strike more quickly with a weapon can considerably increase a character's chance of landing a blow.
A character with the Quick Strike skill always has the option of attempting to attack more quickly than normal. This must be declared at the beginning of the attack, and the player will make a Quick Strike skill check. If the check is successful, the attack takes only half of the normal time, but also causes only 3/4 the regular damage (after all modifiers but before armour effects). If the check fails, the attack takes the normal time, but again causes only 3/4 normal damage.
If the WS is odd, the first such attack will be at half the WS rounded down, and the next will be at half the WS rounded up, so that each pair of quick attacks will take as long as a single regular attack. The damage is always rounded down.
Every weapon handles differently from all other weapons, so this is actually a family of skills, one for each melee weapon. All Quick Strike skills are related to each other.
Quick Strike: Long sword is different from but related to Quick Strike: Club. There is no such thing as Quick Strike: Crossbow, or any other missile weapon.
Note that for characters with Martial Arts or similar skills, the hands and feet (and wings and tail, if applicable) are separate weapons, and require separate Quick Strike skills. There is no requirement that a character with Quick Strike: Hands also learn Quick Strike: Feet.
Brutus has a skill level of 52 in Quick Strike: Short Sword. He decides that he wants to try a Quick Strike, and rolls a 47 on his skill check. This is successful, so Brutus gets an attack with a WS of only 2(=5/2 rounded down). He rolls the attack, and hits. Normal damage for a short sword is d6, but Brutus has an enchanted sword which does d6+2 damage. The damage roll is 2, which would normally cause 4(=2+2) points of damage. Taking 3/4 of this gives a damage of 3 for the attack. He then tries another Quick Strike, but rolls a 76 on the skill check, and has to settle for a normal attack with a WS of 5, again at 3/4 damage. He tries a third and rolls a 29, so this third attack is at a WS of 3(=5/2 rounded up, since the last quick attack was rounded down), still at 3/4 damage.