Leash
A leash is the maximum distance a mob will chase a character before it clears its aggro list and begins to evading attacks. It then will run quickly back to its spawn point (or location it was last patrolling), and reset itself entirely.
In the open world
Leashing is the primary means for players escaping combat.
Leashing is also one way to prevent training, in the sense of griefing somebody by luring mobs onto them. (Since the mob will be in evade mode while returning to its spawn, and will not aggro on anyone it runs past.)
Instances
Non-boss "trash" mobs in instances generally do not leash. The only way to escape them is to exit the instance, or to use an aggro-clearing ability like the hunter's 
[Feign Death]. Some instances however have spots that cannot be pathed to by NPCs. You can identify such a location by attempting to attack an NPC while standing at it; if the NPC faces you but evades, the NPC cannot get to the location. Most of these spots are accessed by jumping onto objects or different elevations that cannot be reached by walking. This is to prevent players from being able to kill NPCs without being attacked themselves.
At these evade locations, a player needs merely to stand there long enough. The mobs will give chase as far as they are able, then stand and stare at you for a while, probably wondering why it is nobody ever taught them to jump, then they will remove you from their aggro list. If the evade-location is within aggro range of their normal patrol route or spawn point though, the leashed NPCs may immediately attempt to re-engage you; although they will be unsuccessful, it will make you unable to leave combat for more than a second or less.
Snapping
When unable to reach a character, certain mobs will snap to the player, instantly teleporting to the player and re-engaging combat with them, ensuring they are not entirely ignored by the party.
Raid bosses
For raid bosses, leashing occurs in a slightly different way. Some boss mobs are intended to be fought in a certain area. If the boss is pulled outside of that area, it will automatically leash, regardless of other factors. Some bosses either do not leash or are fought in an area where players cannot escape, making leashing impossible.
Another example of the leashing mechanic that works in a different way is Iron Juggernaut in Siege of Orgrimmar; if the Iron Juggernaut is pulled outside of the fenced off area towards the friendly NPCs the boss will instantly despawn and the encounter will be reset, causing the boss to respawn about 30 seconds later. This behavior is the most common raid boss leashing behavior.
In Classic
In World of Warcraft: Classic, leashing works differently. Hitting a mob resets its leash range - as such, one way to kite a mob to a far off location is to periodically attack it (for example, walking backwards while meleeing it) to keep its attention.
In Hardcore, this behavior is changed so mobs will not follow between zones.
Raid bosses in Classic
Most of the priests in Zul'Gurub (with the exception of Arlokk the Panther and Bloodlord Mandokir) can be leashed. If a pull is bad or the fight goes awry, it is a common strategy to simply have the remaining people run out of the area. Hakkar will then tell the priest to "not be lured into the mortals' trap," and the boss will reset.
