All weapons have a few different attributes, most important of which are the damage range and the attack speed. Many players look at weapons as a combination of these two factors, the Damage Per Second. However, for many, the attack speed (also sometimes called the delay) is the most important figure.
Literally, the attack speed represents how frequently a character will auto attack with that weapon by default, measured in seconds. There are effects which can increase this such as a quiver, haste rating, or the rogue skill Slice and Dice. Since most weapons are balanced around their Damage Per Second (DPS), weapons with a slow delay will normally have much higher damage values than faster weapons. Whether you are using a slow or a fast weapon should not affect your auto attack damage at all.
However, there are certain abilities that will use the damage of the weapon, such as Sinister Strike. Although many of the common abilities like this have been "normalized" to not make this much of a factor, there are still some that remain that indicate that a player would want a slower weapon. The most recognized of these is Windfury Weapon and Windfury Totem, which gives the shaman and/or his party members an additional attack several times per minute. Because the effect is designed to go off with a certain frequency rather than a certain percentage of attacks, the ideal situation is to have a bonus attack that hits as hard as possible. This means having a slow weapon, since it will have a high damage range and receive a larger bonus from your attack power.
On the other side, there are some abilities and effects which trigger on an attack. Because of this, many players want a high rate of attack so they can create these effects as frequently as possible. For example, while Judgement of Wisdom is active on a target, anyone attacking it has a chance to regain lost mana. With each attack there is a chance to regain mana, so the player would want to strike as many times as possible, and a faster weapon would be superior.
Some other classes have certain sweet spots. Beast Master-spec Hunters ideally use a weapon with a final shot speed of about 1.8 to 2.0 seconds, which means with various haste effects they generally seek a weapon with an attack speed of about 2.70. This allows them to use a special attack between every auto attack, without interruption to either. Faster than this and their auto attacks are delayed. Slower and the frequency of their special attacks is affected.
The one other noteworthy place where attack speed is a factor is in PvP with caster pushback. As a spellcaster completes a spell, a bar on their screen approaches completion. If they are attacked during this time, their spell is partially interrupted and will take an additional portion of a second. While slower attacks do tend to have a greater effect, it is not considerable and players usually prefer to hit casters with a flurry of attacks to limit their spell usage to instant spells only.