When a customer creates a ticket in the Help Desk system, it creates an email thread which is identified by the ticket number in the email subject line (e.g. [#1234] Ticket Created). Updates to that ticket will be emailed to the customer, and if the customer replies to the ticket, their emails will be added to the notes in the ticket.
When a ticket is closed, the customer can still reply to the ticket and their reply will be added to the notes, and if there is an assigned staff person, they will still get the update via email.
However, by default, in that situation, the ticket will remain closed. This is helpful to avoid unnecessary reopening of tickets (e.g. when a customer replies "Thanks for your help!")
Depending on the volume and workflow of your organization's help desk, it may be better to have tickets reopen when a customer replies. There may be occasional unnecessary reopens, but it is usually better to deal with that than to miss a reply from a customer whose issue isn't resolved.
To make this change, go to System > Preferences > Client Preferences. Click the Add button, and select "Reopen Help Desk Entry on Customer Reply." Select True in the Client Setting Dropdown and click Save.