Attack Speed (also known as Attacks), is a stat that determines how often your character will automatically attack in combat.
A player's Attacks stat starts at 20, which can be improved with the multiple attacks skill, widely available at almost any skill teacher. When maxed, your base attacks will be in the low 30s range.
Your attacks stat can be further modified with equipment and weapons. The ATTACKS effect that sometimes appears on equipment directly increases your attacks stat. If you have a base attacks stat of 20 and wear a piece of eq that has the 'ATTACKS by 5' effect, your attacks stat is 25 while wearing the eq. A piece of eq with a negative ATTACKS effect will affect your attacks stat, as well.
A weapon's speed also modifies your Attack Speed as follows:
slowest is -13 and less very slow is -12 to -8 slow is -7 to -3 normal is -2 to 2 fast is 3 to 5 very fast is 6 to 8 fastest is 9 and higher
If the weapon has any ATTACKS effects, they will not be factored into the displayed weapon speed.
The attacks system works like an 'attack timer'. Your attack speed is used to recharge the timer, and when the timer is full, you get to attack. Each attack requires about 300 attacks before the timer is full.
For example, if your multiple attacks skill is untrained (20) and you are using a normal speed weapon (0), your attack timer will recharge in 300/20 = 15 slices, which is a little less than two rounds. (A round is defined as 8 slices.)