Archiving a customer removes them from any contact drop down lists, but retains all their history. Alternatively, if the Customer has no transactions associated with them you can delete them, which removes their details from the system.
Under the appropriate customer click on the Edit contact link.
Click on the Archive button or, if available, the Delete button.
As noted above, the delete button will only be available if there are no transactions (including voided transactions) associated with the customer.
When you now navigate to Contacts > Customers, the customer will appear in the list as Archived.