Attack Speed shows how fast you can attack with your weapon.
Think of attack speed as a global modifier to all of your damage dealt. At 200% attack speed, you can inflict twice as many hits and because of that deal twice as much damage over the same period of time. This is where the Damage Per Second metric comes from.
All weapons have a certain basic attack speed (some variance from the baseline 100% value). This value is modified by any Attack Speed or Total Speed modifiers affecting your character. Be mind about two things. First, your attack speed affects the actual animation time of your attacks, which influences how quickly you can deal damage and then disengage from melee, which is extremely important for ranged characters for one thing, since melee is usually a bad place for them to be. Second, and of greater importance; skills that have a % chance to trigger are NOT scaled to your attack speed.
That is, if you have a 10% chance to deal 50% more damage per hit, that chance is not on a sliding scale 'balanced' to your attack speed. It is ALWAYS 10%. And this means that attack speed can have a much greater impact on your overall damage output than it may first appear.
The maximum attack speed that is achievable with each type of weapon: