An Enrage Timer is an Inferno-specific game mechanic that urges the player to kill or be killed.
After five minutes (clarification needed), an elite will enter an enraged mode. For normal elites, the enrage causes the player to lose health every second until the elite is killed. There is a notification on screen informing the player that the elite has entered enraged mode, and the elite will also have an icon over their head in the shape of a skull and crossbones. In addition, the player will have a debuff icon above their action bar informing them of the damage they are taking every second. This debuff is called Out of Time, and becomes deadly very quickly.
In addition, a boss may have enrage mechanics that aren't displayed as obviously in Inferno difficulty. An example would be that the fight with the Butcher will become increasingly more deadly as the fire grates below the player's feet will activate more frequently as time progresses during the battle. After nearly two minutes, the entire boss arena will be cloaked in flame with no safe place to step. Clarification is needed on this point, as it differs from the Out of Time debuff.
The design intention behind the enrage timer is to not allow a player to endlessly kite an elite monster across a map, but this quickly becomes problematic when some elites spawn with nasty defensive boss modifier combinations such as Invulnerable and Shielding.