DPS is an acronym for Damage Per Second, and can be used in several ways:
Imagine a paladin with a two-handed weapon. Imagine a rogue with a pair of small daggers. Who is going to hit harder? Who is going to hit more frequently? Obviously the paladin, and the rogue, in that order. But who will deal more damage over time? This is where DPS figures in.
Physics tells us that "work equals rate times time." If the work that characters in battle do is damage, then that means damage = rate * time. There's some exact amount of damage you want to deal to ultimately kill a target, but to reduce the time it takes (so you kill it before it kills you,) you need to increase the rate at which the damage is being dealt. This is why every weapon in World of Warcraft has not only the damage range on it and the attack speed, but also the damage per second. The DPS on a weapon is found from those other two statistics, but it is the single easiest way for determining if a weapon is more potent than another. Damage per second for a weapon can be computed as:
(Min Damage + Max Damage) / (2 * attack speed)
A character's DPS is defined as the amount of damage they deal across a certain amount of time. If a player deals 300,000 damage to a boss during 5 minutes (300 seconds), then they would be averaging 1000 damage-per-second. Since different factors can influence how long the battle takes, such as how well the other players in the battle are doing, it is not fair to simply compare a raw amount of damage. On another day, another player might deal 320,000 damage, but take 6 minutes (360 seconds) to do it. Even though they dealt more damage, they had more time available, and were not as efficient during the time the battle took. In fact, they were dealing 320,000/360 = 889 DPS. The first player was about 11% more effective. This is why DPS is the most significant quality for measuring damage output.
Players who are dealing damage (see below) should be focused on figuring out how to deal as much as possible. They still want to consider factors like threat and the cost to use a particular ability, since a sudden burst of damage might not be very helpful in the long run, but they should be doing what they can to increase it. Especially during raiding, DPS classes will do whatever they can to maximize their damage, including tweaking their gear and enchants, using consumables, and exploiting group synergy.
Most characters will want to focus on the abilities that also have the best return on damage over time. The measurement for this is similiar -- damage dealt divided by time to create that damage. This is not the only characteristic players use for evaluating abilities; they might also want to consider DPM (damage per mana) to see if the ability is cost-efficient, or they might use a skill for some secondary benefit it provides.
On the character window, there are two panes in the center beneath the actual character. If one of these is set to melee or ranged, there is a line for "Damage." Mousing over this will report back a Damage Per Second value for autoattacking with the character's weapons. Unless you are considering a wand, that DPS will not match the printed DPS of the weapon because other factors increase the damage your character deals with it. For example, attack power increases DPS by 1 for every 14 points of AP. Many other factors contribute to changing this number, including haste, talents, buffs, and other effects. This DPS value is also not the same as the damage output of your character, because many other factors change the damage dealt, such as armor and crits. Also, each character will be using abilities. In the case of some classes, their damage will come entirely from their abilities and not from their weapons.
Characters who tend to deal damage during a group are often called "DPS". Unlike a tank or a healer, they have no real function other than to dispatch enemies. Some DPS characters might also be expected to provide utility, crowd control, or to pull, but they will probably not devote their entire time to these tasks and will deal damage in-between.
As a general rule, Rogues, Hunters, Mages and Warlocks are universally regarded as DPS classes. The other classes that can heal or tank all have a way of configuring their talents to deal damage if that is the way they choose to play their character. They may or may not be less effective than the "pure" DPS classes, but some specs are very potent. Many DPS specs for these other classes are considered very helpful to the group as a whole, and should not be ignored. (Shadow priests, in particular, have made a name for themselves during Burning Crusade.)
The role of DPS is as simple as dealing as much damage as possible without causing harm to themselves or others because of it. If a damage dealer can be sure to not rip aggro or to break crowd control, simply pushing themselves to the limits will be, in most cases, a job well done. Intricate raid fights usually require DPS players to monitor more than their threat meters.
Raid leaders will often use DPS as an action to mean "dealing damage" when they want to influence the rate damage is being dealt.
DPS harder! We need to kill him before he enrages.
Stop DPSing until after this phase is over.