Damage per second (DPS) is a measure of damage dealt over one second. DPS is a more practical and comparable measure of damage output than plain damage.
The term DPS is also sometimes used figuratively to describe the act of applying damage to a target over time (as in "all DPS on the boss" or "stop all DPS"), or to refer to a class whose primary role in a group is dealing damage (as in "Arcatraz group needs one more DPS to roll").
Burst DPS
Burst DPS refers to DPS dealt over a relatively short period of time. High burst damage is the preferable form of damage against targets with relatively low health.
Mages can deal high burst DPS due to the high damage output of their spells, specifically the Arcanist but lack the means to maintain it for an extended period of time.
Dagger-wielding rogues have the ability to use 
[Ambush], which is also considered burst damage due to its ability to produce a single very high damage shot directly from stealth.
Enhancement shaman are well-known for their burst capacity du to the windfury procs, completed by Shock spells.
Sustained DPS
Sustained DPS refers to DPS dealt over an extended period of time. High sustained damage is the preferable form of damage against targets with relatively high health.
Rogues can deal high sustained DPS because they can keep fighting as long as they have health; their energy regenerates very quickly.
Shamans can deal both burst and medium sustained DPS, an enhancemnet shaman can Dual wield, a fast offhand weapon = keeping Flurry and Unleashed Rage up with offhand crits + windfury every 5 seconds = Good sustained DPS. On the other hand, a slow 2 handed weapon provides very good Burst DPS if windfury procs.
Warriors are also adept at sustained DPS due to the rage system which allows them to use more of their abilities as they gain more rage, which accumulates from both taking and dealing damage. Because of this, warriors become more deadly as fights drag out, whereas other classes who are reliant on mana, such as mages and priests, can't do much of anything once their mana runs out.
Hunters have decent DPS, as even when their mana runs low, they do not have to rely on secondary attacks to continue DPSing. This is in constrast to primary casters, who need to switch to wands (a secondary attack which usually has a much lower DPS rate than casting) to deal damage when out of mana.
Warlocks also have very good Sustained DPS, thanks to their DOTs, Burst Damage Spells, and 
[Life Tap], an ability that allows them to exchange health for mana, thus allowing warlocks to ensure the process.
Weapon DPS
Weapons will often be quoted along with a DPS figure. This provides a useful means of comparing weapons; however, your personal DPS using that weapon will vary hugely depending on your level, your stats, the mobs you are fighting, and many other factors.
Spell DPS
Some sites will quote the DPS for a spell. This may be calculated as Damage / (Cast Time + Recast Time) or just as (Damage / Cast Time) so beware of the difference. DPS monitoring tools sometimes struggle with spells due to their one-hit nature, and the ability to cast combinations of spells.
Calculation of DPS
To calculate the DPS on a weapon you need two things, and that is Speed and Damage. Here's an example:
Gladiator's Painsaw, 201 - 303 Damage, 2.20 Speed
And how to calculate this: (201 + 303)/2 = 252
This is how much damage the weapon averages per hit. To figure the average damage per second, divide it with 2.2: 252/2.2 = 114.5454...
This would be rounded to 114.5 DPS.
The information delivered in your tooltip from your paper doll window does not take into account hit rate. Some addons do, and may display the total average DPS in one tooltip or another.
Additional notes
- UI modifications such as Cosmos include tools to enable players to track their DPS.