Snapshot
In some MMORPGs, a "snapshot" is the point in time when the calculation of an instance is completed by the game's servers. There is no official term for snapshotting. In Final Fantasy 14, there are numerous examples of when snapshots may occur, whether it is performed by the player, or enemies.
Player Damage
The most common example of snapshotting, a snapshot occurs when an ability locks in (snapshots) the player's stats and buffs at the moment of execution. Actions in PvE will almost always immediately snapshot by the moment of execution, meaning that if an action's damage pop-up appears delayed, its damage has still been snapshotted. An example of this is with Ouroboros, a 
viper action where the damage appears to apply late at the apex of the animation, but its damage has still been snapshotted immediately upon execution.
All damage over-time effects are also snapshotted immediately when applied, and instances of damage dealt over-time also deal damage based on the player's stats at the time of execution, even after the expiration of buffs or stat changes. See examples here. Jobs with access to actions that apply damage over-time effects are encouraged to refresh their duration just before raid buffs expire, even if it may refresh the timer early. This is especially important for 
healers and 
bards.
Damage Taken
Another common instance of snapshotting is when taking damage. Any status effect effects that may increase or decrease the damage a player is snapshotted upon mechanics actively resolving, typically after a cast bar finishes.
A snapshot occurs by bosses when the cast bar for an action they are casting is completed. Players must use any mitigation actions before the cast bar completes, or else they will have no effect on the damage dealt when the boss's respective attack animation completes. Because enemies cannot perform auto-attacks while casting, players will often use their mitigation tools near the end of the cast bar. If timed correctly, the mitigation effects may resolve further instances of damage. An example is a 
tank applying Rampart to themselves just before a tankbuster's cast bar is complete, allowing the damage-mitigation effect of Rampart to mitigate damage from not only the tankbuster itself, but subsequent auto-attacks thereafter.
Another example is a 
samurai taking damage while under the effect Third Eye or Tengentsu. These actions are unique in that extra Kenki and any additional effects will only be applied when taking damage. Because of its short duration, these actions should be executed before the end of a boss's cast bar. If the duration of either action expires after the cast bar finishes, the Kenki and any additional effects will still be granted to the samurai upon the damage pop-up appearing.
Furthermore, damage from mechanics are snapshotted when the cast bar or when area of effect indicators (often orange circles) disappear, and not when the animation finishes.
Examples
Biolysis (PvP) deals 3,000 potency over-time, and Guard (PvP) grants damage mitigation of 99%.
PvP actions are chosen as examples due to the flat amounts of potency, with no damage variation.
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An example of the full amount of damage being dealt, even when the target is mitigated.
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An example of damage is snapshotted when mitigated, dealing little damage even after the target is no longer under any mitigation effects.