Enmity (also known as hate) is Final Fantasy XI's term for how much a monster hates any particular player. This is the quantity which determines whom a monster would be attacking during battle, and is comparable to terms like Threat Level in some other MMORPGs.
Enmity is not to be confused with aggression, which means that a monster detects a player and starts attacking. To be able to gain enmity, a player has to be on the monster's enmity list, a list of players toward whom it holds hate. To get on that list, a player either has to perform any kind of action on the monster itself (i.e. cast a spell on it, or attack it), perform an action on someone who is already on the monster's enmity list (i.e. cure or buff the player), or simply rest near a monster which has established enmity on a party or alliance member.
Player testing indicates that the total amount of enmity for a player consists of two components, a static form of enmity (referred to as cumulative enmity or CE), and a dynamic form of enmity (volatile enmity or VE). The difference between the two is that cumulative enmity stays at the same level and can only be lost by taking damage, while volatile enmity will decay over time, whether any damage is taken or not. So far, almost every single action seems capable of producing at least a small amount of cumulative enmity, and many actions seem to be able to produce both cumulative and volatile enmity. The amount of enmity produced however may differ greatly.
Upon dying, a player will be removed from all monsters' hate lists it was on before. That means the monster(s) won't pursue this player anymore. Zoning will also work but only if no other players remain on the monter's hate list.
There are a number of possible situations that might arise from this:
Enmity gain can be modified with certain equipment and merit points. Enmity +/-x means a percentual increase/decrease in total enmity gain.
The concept of shared enmity or shared hate has nothing to do with actual enmity, as described in this article. It's a colloquial term that describes aggression and fighting behaviour of certain monsters or monster families. It differs from enmity in that it is gained by killing monsters of a certain type, and that every monster of the family will act according to that number. Depending on the actual monster family, it can either be used to modify aggression behaviour (as with Fomors and possibly Pixies), or even affect the battle style and single attacks (i.e. Apkallu and Tonberries). While it still may have effects on enmity gain, depending on the monster family, it is coincidental and unrelated to enmity as mentioned here. Usually it is possible to reduce accumulated shared enmity, but that also differs with each monster family.
Almost every party or alliance must manage their players' respective enmity levels in some way. Usually the focus is to keep the hate on the tank, the person with the highest chance of surviving the enemy attacks. However, some party setups with no real tanks just rely on the melees constantly out damaging each other, so that tanks are switched regularly, which gives enough time to recast shadows, or to bind/kite the mob around while nuking, which is used in an all BLM setup.
Some mobs behave differently in certain situations involving enmity.
The special move Spike Flail for instance, native to a variety of wyrms, is triggered when a player standing behind the wyrm in question suddenly gets hate. Since this move usually results in the death of most, if not all members of the alliance, people try to avoid it at all costs. But since players will get hit by cone damage attacks like breaths if they stand in front of wyrms, they have to be very careful where to stand, and to control their enmity gain if they're behind the wyrm.
A few moves are known to reduce or reset enmity. These can either be single target (like Delver's Impalement, or Vrtra's Horrid Roar) or target everyone in the area of effect, with range differing greatly (i.e. Kam'lanaut's short ranged Great Wheel, or Sand Trap, a move native to the Antica family). These can prove to be a threat, since the tank needs to get hate back fast, or the monster will target the next person who uses a weapon skill or a high enmity ability. And since mages have and use high enmity spells, they risk being attacked by the monster if the timing is bad and the tank is out of means to generate enmity fast. Also, in the case of a short range enmity resetting move such as Sand Trap, mages are usually out of range of the move when it occurs, thus their enmity is not reset at the same time as the front-liners, and it becomes harder to regain enmity from the mages if they have not managed their enmity gain to some degree.