A Persistent Instance is one that does not release when the last player leaves it. Its state is preserved for a number of days and, if anyone in the original party re-enters that instance, the instance is restored to its prior state.
This means that a raid no longer needs to complete an instance in a single, marathon session. They can set a time limit for the first session and meet again the next day to continue where they left off. This will also remove the issue of a server reset during a raid.
Persistence is shown as a range of days. The minimum value is the same as the Success Lockout. This is the minimum time that must pass before you can release the instance you started and initiate a new instance. The maximum value is the longest period you can extend the instance over.
For example, Emerald Halls has a persistence of 5-9 days. This means that once you start a run in Emerald Halls, you cannot release and restart Emerald Halls for 5 days. After 5 days you will have the choice to release the instance or re-enter the copy you already have. You can extend the instance to a max of 9 days, at which time the instance would vanish, kicking out all players currently inside, and you must start over in a new copy of the instance.
This feature went live with LU37 on July 31, 2007.