A burn or burn phase is a period of time during which one or more characters in a party attempt to deal as much damage as possible. This usually involves activating a string of offensive abilities and disciplines with long cooldowns, receiving temporary buffs from support classes, and/or using powerful but mana-inefficient direct damage spells.
Wizards and berserkers are specialists at burn damage, although all classes (particularly DPS classes) possess at least a few abilities for burning.
In raiding, burning is often tactically important for quickly finishing a phase of a boss fight during which the boss's attacks or other fight mechanics are especially dangerous.
In soloing, moloing, and grouping, burning has applications in defeating difficult named mobs in a hurry, mustering enough damage to get through particularly difficult sections of a mission, or simply increasing experience gain by optimizing how fast foes die.