Damage per second (DPS or dps) is a measure of the damage dealt by a person or group over one second. DPS is a more practical measure of damage output than plain damage, as it allows characters of differing levels and classes to effectively compare their damage output.
The term dps is also sometimes used to describe the act of applying damage to a target (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 "Tutu group needs one more dps"), i.e. damage dealer.
"Burst DPS" refers to damage dealt over a relatively short period of time. High burst damage is the preferable form of damage against targets at/with relatively low health. Most classes have at least a few spells or abilities that generate high damage very quickly. This type of damage is preferred in PvP environments, but is usually too expensive or has too long a cooldown to be sustainable.
"Sustained DPS" refers to constant damage dealt over an extended period of time. High sustained damage is the preferable form of damage against targets with relatively high health. Every class is capable, when specced and geared properly, of doing good sustained damage. Acolytes tend to have the highest sustained dps, as their most damaging attack takes no MP.
Sometimes spells are referred to in terms of their dps and is called Spell DPS. This may be calculated by Damage / (Cast Time + Recast Time) or just as (Damage / Cast Time) so be aware of the difference. In contrast to this, spells with higher dps often tend to be lower in DPM (damage per mana).