Burst damage is a term used to describe dealing high amounts of damage in a very short period of time. Spells which qualify for burst damage are often mana inefficient. Burst damage may also refer to some PvE encounters in which a boss will inflict a lot of damage in a short time.
Doing burst damage normally causes the player to run out of mana much faster. In PvE, it also gives the opportunity to draw aggro, making it inefficient and even dangerous as a means of beating bosses in most cases. However, in some encounters, burst damage is necessary. Its main purpose in PvP is to bring an enemy down fast, often combined with Focus Fire, giving enemy healers as little chance as possible to heal the damage inflicted.
Abilities that can be used to do burst damage normally cause unusually high damage quickly (for example, Mind Blast and most instant cast direct damage spells) or extraordinarly high damage (e.g. Pyroblast and Soul Fire). Talents that allow burst damage often increase the damage caused (Arcane Power, Power Infusion) or give the opportunity to cast a spell instantly (Presence of Mind, Nature's Swiftness).
Certain classes are better at burst than others, and certain specs of these classes. Some of the best are: