By default, you will receive enquiries sent through the LiveZilla web interface (the chat window your customers see) only. If you want to receive incoming emails in LiveZilla, please follow these steps.

If you have changed this previously, please make sure that the ticket system is enabled under:
Server Configuration -> Tickets

Configure mailboxes for incoming (IMAP or POP) and outgoing (SMTP) emails under:
Server Configuration -> Emails

Configure your groups accordingly under:
User Management -> Groups -> Group -> Tickets

Configure Ticket permissions under:
User Management -> Roles -> Tickets

To send emails with HTML you need to configure a HTML template first. In order to do so please go to:
User Management -> Groups -> Group -> Languages -> Language -> Emails