On the customer portal, your customers will be able to use the New Support Ticket option to submit tickets from the portal. This option can be made available to either all users or logged-in users only as well, by making changes to the User Permission section under Admin -> Channels -> Portal -> Settings. This article has detailed information on setting it up.
The fields that appear on the ticket creation form on the customer portal can be configured under Admin -> Workflows -> Ticket fields.