Attack Speed determines the frequency at which you can use your Basic Attacks, the value is equal to the attacks per second, it maxes out at 3. You may move during the down time between Basic Attacks or use abilities or items, the timer will continue to count down. Below is a list of all artifacts giving Attack Speed, the amount granted at max Tier, and the cost at max Tier.

Max attack speed is a value of 3 also being 3 Basic Attacks/second.