Enmity is Final Fantasy XI's term for what was once commonly called "hate", or how much aggression a particular monster has for any particular player (i.e. how much the monster wants to kill any given 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. (See Dictionary Definition.)
Player testing indicated that Enmity has two components, Cumulative Enmity (CE) and Volatile Enmity (VE). The difference between the two is that Cumulative Enmity does not decay over time, while Volatile Enmity does. The total of two quantities is Total Enmity (TE, or just Enmity), which dictates the target of the monster. So far, almost every single action seems capable of producing at least one unit of CE, and many (if not most) actions seem able to produce potentially both CE and VE.
"Hate" is the older conventional term used by many English speaking FFXI players to describe Enmity. Note that Square Enix does not use the "hate" within game or when the company describe monsters' behavior and aggression. For that, the company uses Enmity.
Also, the player spearheading the Enmity testing prefers Enmity, Total Enmity, Cumulative Enmity, and Volatile Enmity, so familiarity with those terms will aide the understanding of the most authoritative references in existence.