If a customer decides to cancel and then changes their mind, you can reactivate their subscription if the subscription has not yet expired. Once the subscription has expired, your only option it to start a new subscription on the customer's account.
Reactivating a subscription will move the subscription back to an active state and will not issue an invoice to the customer. Reactivating the subscription will ensure that the subscription renews at the end of the current cycle.
If you are allowing the customer to cancel their subscription from the Hosted Account Management pages, the customer will see the option to reactivate in place of the cancel link until the subscription expires.