persistent session
A connection where the broker remembers the client's subscriptions and saves QoS 1 and 2 messages for it while it is offline.
A persistent session (clean session off) is kept by the broker under the client ID across disconnects: its subscriptions survive and QoS 1/2 messages that match them are queued and delivered when the client reconnects.
Examples
A pump controller on a persistent session gets the QoS 1 commands published while it was rebooting once it reconnects with the same client ID.
Mosquitto keeps up to 1,000 queued messages per offline client by default and drops the rest.