The Remember Subscriber option/feature allows subscribers to claim new offers from a directory that they're already subscribed to. They can also reclaim renewable offers from the same directory.
This option is found in Campaign Properties (Directory) > Directory Settings
Each option indicates how the remember feature works for each subscriber.
For instance, "daily" lets the directory remember each subscriber on a daily basis. If a user subscribes today, they'll be subscribed for this day, but not tomorrow. When they sign up tomorrow, it'll be a completely new subscription. If the user subscribed again to the campaign today while the Subscriber Duplication is disabled, they won't be able to subscribe to the same campaign again.
On the other hand, "weekly" allows the directory to remember each subscriber for a week. So each subscriber will be subscribed to the directory for 7 days / 1 week. Each subscription they make before the 7th day since they subscribed will be considered duplicate subscriptions.
The same thing for the other timeframe except for Infinite.