Berserk
Berserk is a mechanic that sharply increases the damage output of most[1] bosses after a certain time (typically by 1000 %). This time limit is shown as a timer (on the right side of your screen by default) when you engage the boss in battle. You need to defeat the boss before this timer runs out, as typically you will not survive much longer than a few more seconds after the boss goes berserk.
On most bosses you will receive a warning between thirty seconds and one minute before berserk. If the timer does run out, you will normally receive another warning to alert you that the boss is entering a "frenzied rage"[2]. Seeing this message is an indication that the damage your group deals was too low - so try to identify how you can increase it. A good way to do this is using the "battle stats window" (hotkey B), or the combat log.
Some bosses use the berserk mechanic to prevent exploits. For example, if Ragefang or Goldsnare are kited to the respawn point of the spiders, they will immediately go berserk, regardless of how much time was left.
Calculation
Berserk timers are set so that a reasonably equipped group can defeat the boss with 30-60 seconds to spare. For groups with more than three players, the damage required for each player is adjusted so that one player (if engaged by four players) / two players (if engaged by five players) can be knocked out for 25% of the time.
All calculations assume that the group has one tank, one healer, and the rest playing as damage dealers.