Burst damage or burst potential refers to a player's ability to deal a large amount of damage in one or two turns, allowing a player to finish off a previously damaged opponent, or deal an overwhelming amount of damage that the opponent cannot quickly recover from. It can play a role both in aggro decks, which may need burst damage to finish off the weakened opponent if the opponent is able to win back the tempo, and in control decks, which may use burst damage as the primary mechanism for killing the opponent after playing defensively for most of the game.
Burst potential stands in contrast to repeated damage done by minions while controlling the board. While such minions can do significant damage simply by staying in play and attacking over many turns, burst damage is characterized by its use of one-time effects played from the hand to generate a surge in damage output, possibly in combination with minions and weapons already in play.
A one turn kill (OTK) is a concept closely related to burst damage, describing defeating the opponent in a single turn, generally as a result of strong burst damage. The exact definition of a one turn kill varies: for some it requires taking the opponent from 30 to 0 Health in a single turn, with no cards previously played, while for others it simply requires dealing a large and lethal amount of damage in one turn.
Uses
Large amounts of burst damage often are used to damage the opponent's hero rather than minions, for several reasons. Foremost is that by the time a player has the resources to launch such an onslaught of damage, the opponent may already be close enough to dead to finish them off. Or, they may be brought low enough that the player expects to be able to force through a victory shortly thereafter. Another is that the damage combination might account for so much of the damage-dealing potential of the deck that the player is unlikely to be able to actually kill the opponent without it. Furthermore, depending on the specific combination used, it may be difficult to distribute the damage among enemy minions without wasting much of its potential, since most minions don't have double-digit health. Also, if minions produce the damage burst, hitting the enemy player is "free" in that the minions take no damage in the process, remaining as threats the opponent must address - perhaps by trading their own minions anyway. However, some high burst damage methods are more flexible and may be used efficiently to board control the enemy, depending on the situation.
Smaller amounts of unexpected damage can be highly valuable in making favorable trades. These usages are not what is primarily thought of as "burst damage", but technically even the smallest attack buff that was not visible on the board beforehand could be called "burst", since it is above and beyond the amount the opponent knew to expect based solely on known minions.
Sources
Burst may be generated from spells, minions with Charge and damaging abilities, and combinations of cards that increase each other's effectiveness. Mana acceleration and Spell Damage are common ways to enhance burst power.
Combos
While there are some neutral minions capable of contributing to burst damage, such as 
Leeroy Jenkins, the most powerful combinations tend to require some class-specific spells, weapons, and minion abilities that synergize for as much damage as possible. Specific burst combinations therefore vary widely between classes. The following list suggests one or two typical strategies each, but it is far from comprehensive:
- Mages may simply use a series of direct damage spells Fireball and Pyroblast to deal high damage.
Frostbolt and Ice Lance are effective methods in Wild format with cost reduction and Spell Damage. - Priests have multiple methods to deal burst damage. They can employ Divine Spirit to boost a minion's Health to extreme levels then follow up with Inner Fire to convert that into massive damage. They can also use Mind Blast with Prophet Velen for massive damage, which often required Resurrection effects to summon Velen at a lower cost. Shadowreaper Anduin's Voidform can be comboed with cheap cards to maximize Hero Power usage. Shadow Visions is extremely helpful for searching for combo pieces and Radiant Elemental allows for more card plays per turn.
- Warriors can Enrage a charging Grommash Hellscream for strong burst, usually as a finishing move. If it's Recruited with a Woecleaver, he can be copied with Sudden Genesis for even more damage.
- Rogues have spells like Preparation and Shadowstep to enable more damaging spells and charging minions to be played in a turn than would otherwise be possible, plus Cold Blood for a large Attack boost.
- Shamans have the ability to apply Rockbiter Weapons and/or Flametongue Totem to characters with Windfury, or use Bloodlust to turn a field of seemingly weak minions into massive damage. Overload allows shaman to defer the cost of spells like Lava Burst, Crackle, and Lightning Bolt to later turns, making them good sources of rapid damage as well.
- Hunters can employ Unleash the Hounds together with other beast synergies like Timber Wolf, Leokk, and the cheap Kill Command to produce quite a lot of damage. King Krush is an alternative big damage finisher for the late game.
- Warlocks can summon a Doomguard for hefty charge damage, which can be copied with Carnivorous Cube then killed with Dark Pact to triple the damage. Soulfire is another option for cheap damage, and in Wild format, Power Overwhelming on a charge minion is another method of dealing high damage for a low cost.
- Paladins tend to depend on board control over burst damage. However, their Blessing of Might can be used in similar combinations as Power Overwhelming, while Blessed Champion and Dinosize serves as another way to give their minions a sudden burst of damage.
- Druids have limited methods of dealing burst damage, but if they can maintain a wide board, their AoE buffs like Mark of the Wild, Savage Roar, Cenarius, and Branching Paths can turn many weak minions into many decent sized minions.
Previous combos
The following combos are no longer possible due to card changes.
- Prior to Patch 10.2.0.23180(2018-02-06), Raza the Chained makes Hero Powers cost (0). Combined with Voidform and cards typically Prophet Velen, Radiant Elemental, Mind Blast, Holy Smite and Potion of Madness, Highlander Priests can deal up to 38 damage in one turn.
- Prior to Patch 6.1.3.14830 (2016-10-03) Charge's text was "Give a friendly minion +2 Attack and Charge." Strong example uses were Raging Worgen and Frothing Berserker, both of which when used in combination with Whirlwind or similar effects can increase their power dramatically.
- Prior to Patch 5.0.0.12574 (2016-04-24), the Treants summoned by Force of Nature had Charge. Combining Force of Nature with Savage Roar was a very popular druid combo, frequently used as a OTK.
Counters
Killing minions likely to be used in burst combinations is always a good idea, although damage played entirely from the hand is of course immune to this board control strategy, forcing opponents to address its threat preemptively rather than reactively. Other defenses against burst damage include Taunt minions to stop Charge, prevention of spellcasting with 
Loatheb or Counterspell, and simply keeping life totals too high to die in one turn, possibly using healing or Armor effects. Keeping the opponent on the defensive, thereby forcing them to use up their damage spells on minions, may also pressure them out of accumulating enough cards to play a successful burst combination.