Rage is when a monster has taken enough Physical Damage and reacts by becoming much more aggressive. Many monsters signal that they are entering an enraged state by roaring but some, like Yian Kut Ku and Gypceros, jump up and down. Those monsters that don't display still have signs that they are enraged, such as markings and/or huffs of smoke coming out their mouths. The attack, defense and agility for each monster significantly increases (to the point that some will one hit KO enemies), as well as certain tolerances (in the case of Diablos and Monoblos, they become unaffected by Sonic Bombs while enraged). It's even possible to use Pitfall Traps on an enraged Rajang or Nargacuga, who normally avoid Pitfall Traps.
How much the stats are increased depends on the monster. Monsters usually get enraged more often when they are close to death, a famous example being the Diablos. Rage can also be induced by waking certain sleeping monsters, and in the case of Yian Kut-Ku, by stunning it with a Sonic Bomb or explosive, or occasionally by breaking a body part (a Shogun Ceanataur becomes 'locked' into Rage when one of its claws is broken).
Most of the time, if a monster is in the middle of an attack and a hunter drives them into rage, they will stop that attack and do the Rage state indication.
Some monsters, such as a Khezu or a Tigrex, sometimes go into a state of "constant rage", in which once that monster is out of rage state, one hit could set it off again.